Home Injective and coherent endomorphism rings relative to some matrices
Article Open Access

Injective and coherent endomorphism rings relative to some matrices

  • Yuedi Zeng EMAIL logo
Published/Copyright: September 4, 2023
Download Article (PDF)
Open Mathematics
From the journal Open Mathematics Volume 21 Issue 1

Abstract

Let M be a right R -module with S = End ( M R ) . Given two cardinal numbers α and β and a row-finite matrix A ∈ RFM β × α ( S ) , S M is called injective relative to A if every left S -homomorphism from S ( β ) A to M extends to one from S ( α ) to M . It is shown that S M is injective relative to A if and only if the right R -module M β ∕ A M α is cogenerated by M . S is called left coherent relative to A ∈ S β × α if Ker ( S ( β ) S → S ( β ) S A ) is finitely generated. It is shown that S is left coherent relative to A if and only if M n ∕ A M α has an add ( M ) -preenvelope. As applications, we obtain the necessary and sufficient conditions under which M n ∕ A M α has an add ( M ) -preenvelope, which is monic (resp., epic, having the unique mapping property). New characterizations of left n -semihereditary rings and von Neumann regular rings are given.

Keywords: coherent; injective; endomorphism ring; preenvelope; quasi-projective
MSC 2010: 16D40; 16D50; 16P70

1 Introduction

Throughout this article, R is an associative ring with identity, and all modules are unitary. Let M be a right R -module with the endomorphism ring S = End ( M R ) . Obviously, End ( M ) M R is right balanced. In [1], Garcia et al. investigated the FP-injectivity and coherence of S . In [2,3], Mao investigated the ( m , n ) -injectivity, ( m , n ) -flatness, and ( m , n ) -coherence of S . Motivated by [4–9], we shall consider the injectivity of M and the coherence of S with respect to certain matrices.

Let M be a right R -module with S = End ( M R ) and A ∈ RFM β × α ( S ) . In Section 2, we say S M is injective relative to A if every left S -homomorphism from S ( β ) A to M extends to one from S ( α ) to M . And S M is said to be ( α , β ) -injective if it is injective relative to all A ∈ RFM β × α ( S ) . So, a united frame is provided for investigating P-injective, finitely injective, ( m , n ) -injective, FP-injective, and injective modules over its endomorphism ring (see [1–3, 10–12]). It is shown that S M is injective relative to A if and only if the right R -module M β ∕ A M α is cogenerated by M . We also extend the notion of quasi-projectivity to quasi-projectivity relative to A . It is shown that if S M is injective relative to A (the condition is necessary), then the following conditions are equivalent:

  1. S is left injective relative to A ( S is called left injective relative to A [9] in case, for every h ∈ Hom S ( S ( β ) A , S ) , there exists g ∈ Hom S ( S ( α ) , S ) such that h = g η , where η : S ( β ) A → S ( α ) is the canonical inclusion).

  2. σ ∈ A S α , for any right R -morphism σ : M → M β with Im σ ⊆ A M α .

  3. M is quasi-projective relative to A ( M is called quasi-projective relative to A in case, for every h ∈ Hom R ( M , A M α ) , there exists g ∈ Hom R ( M , M α ) such that h = π g , π : M R α → A M R α ).

As a consequence, we prove that if M is endo ( m , n ) -injective, then S is a left ( m , n ) -injective ring if and only if M is endoquasi- ( m , n ) -projective. We obtain a new necessary and sufficient condition of left ( m , n ) -injective rings in [3, Theorem 2.4].

Recall that a ring R is said to be left coherent [13] in case each finitely generated left ideal of R is finitely presented, or equivalently, any finitely generated submodule of any finitely generated free left R -module is finitely presented. Let M be a right R -module with S = End ( M R ) and A ∈ RFM β × α ( S ) . In Section 3, we say S is left coherent relative to A if Ker ( S ( β ) S → S ( β ) S A ) is finitely generated. It is shown that S is left coherent relative to A if and only if M n ∕ A M α has an add ( M ) -preenvelope. Next, we investigate the necessary and sufficient conditions for monic (epic, having the unique mapping property) add ( M ) -preenvelopes of M n ∕ A M α . As applications, we obtain new characterizations of left n -semihereditary rings and von Neumann regular rings.

Let α ≥ 1 and β ≥ 1 be two fixed cardinal numbers, RFM β × α ( S ) stands for the set of all β × α row-finite matrices over S . We write S ( β ) to indicate the direct sum of β copies of S and S α to indicate the direct product of α copies of S . Elements in S ( β ) are regarded as “row vector,” elements in S α are regarded as “column vector.” For the right R -module M , elements in M ( β ) ( M α ) have similar meanings. Thus, we define right R -homomorphisms θ : M β → M and σ : M → M β as follows:

θ ( x 1 x 2 ⋮ x β ) = ( θ 1 , θ 2 , … , θ β ) x 1 x 2 ⋮ x β = θ 1 x 1 + θ 2 x 2 + ⋯ + θ β x β

and

σ ( x ) = σ 1 ( x ) σ 2 ( x ) ⋮ σ β ( x ) ,

where x ∈ M , x 1 x 2 ⋮ x β ∈ M β , θ = ( θ 1 , θ 2 , … , θ β ) ∈ S ( β ) and σ = σ 1 σ 2 ⋮ σ β ∈ S β .

If A = σ 11 σ 12 ⋯ σ 1 α σ 21 σ 22 ⋯ σ 2 α ⋮ ⋮ ⋮ ⋮ σ β 1 σ β 2 ⋯ σ β α ∈ RFM β × α ( S ) , we can define a right R -homomorphism f : M α → M β ( A M α ) via

f ( x 1 x 2 ⋮ x α ) = A x 1 x 2 ⋮ x α = σ 11 ( x 1 ) + σ 12 ( x 2 ) + ⋯ + σ 1 α ( x α ) σ 21 ( x 1 ) + σ 22 ( x 2 ) + ⋯ + σ 2 α ( x α ) ⋮ σ β 1 ( x 1 ) + σ β 2 ( x 2 ) + + ⋯ + σ β α ( x α ) ,

for any x 1 x 2 ⋮ x α ∈ M α .

2 Injectivity relative to some matrices

Let α ≥ 1 and β ≥ 1 be two fixed cardinal numbers.

Definition 2.1

Let M be a right R -module with S = End ( M R ) and A ∈ RFM β × α ( S ) . S M is called injective relative to A if every left S -homomorphism from S ( β ) A to M extends to one from S ( α ) to M , i.e., the following diagram commutes

where η : S ( β ) A → S ( α ) is the canonical inclusion.

Theorem 2.2

Let M be a right R-module with S = E n d ( M R ) and A ∈ R F M β × α ( S ) , the following conditions are equivalent.

  1. S M is injective relative to A .

  2. E x t S 1 ( S ( α ) ∕ S ( β ) A , M ) = 0 .

  3. σ : M α ∕ r M α ( A ) → Hom S ( S ( β ) A , M ) defined via

    σ ( x + r M α ( A ) ) ( b A ) = b A x , ( ∀ x ∈ M α , b ∈ S ( β ) )

    is an isomorphism of abelian groups.

  4. σ : A M α → Hom S ( S ( β ) A , M ) defined via

    σ ( A x ) ( b A ) = b A x , ( ∀ x ∈ M α , b ∈ S ( β ) )

    is an isomorphism of abelian groups.

  5. r M β l S ( β ) ( A ) = A M α .

  6. The right R-module M β ∕ A M α is cogenerated by M.

Proof

(1) ⇔ (2) is trivial.

(2) ⇔ (3) Define a right R -homomorphism φ : r M α ( A ) → Hom S ( S ( α ) ∕ S ( β ) A , M ) via φ ( u ) ( s + S ( β ) A ) = s u ( ∀ u ∈ r M α ( A ) , s ∈ S ( α ) ). Clearly, φ is well defined. If f ∈ Hom S ( S ( α ) ∕ S ( β ) A , M ) , then there is an element u ∈ r M α ( A ) such that f ( s + S ( β ) A ) = s u for any s + S ( β ) A ∈ S ( α ) ∕ S ( β ) A . Thus, φ is epic. So, ( 2 ) ⇔ ( 3 ) follows from the following commutative diagram with exact rows:

(3) ⇔ (4) Define a right R -homomorphism ψ : M α ∕ r M α ( A ) → A M α , via ψ ( x + r M α ( A ) ) = A x ( ∀ x ∈ M α ). Obviously, ψ is well defined. It is easy to check that ψ is an isomorphism.

(5) ⇒ (4) Suppose A = σ 11 σ 12 ⋯ σ 1 α σ 21 σ 22 ⋯ σ 2 α ⋮ ⋮ ⋮ ⋮ σ β 1 σ β 2 ⋯ σ β α ∈ RFM β × α ( S ) and the t -th row of A is σ t = ( σ t 1 , σ t 2 , … , σ t α ) ∈ S ( α ) , t = 1 , 2 , … , β . Let f : S ( β ) A → M be any left S -homomorphism and y = f ( σ 1 ) f ( σ 2 ) ⋮ f ( σ β ) ∈ M β . It follows that f ( b A ) = f ( b 1 σ 1 + b 2 σ 2 + ⋯ + b β σ β ) = b 1 f ( σ 1 ) + b 2 f ( σ 2 ) + ⋯ + b β f ( σ β ) = b y , for any b = ( b 1 , b 2 , … , b β ) ∈ S ( β ) . If t = ( t 1 , t 2 , … , t β ) ∈ l S ( β ) ( A ) , then t y = t 1 f ( σ 1 ) + t 2 f ( σ 2 ) + ⋯ + t β f ( σ β ) = f ( t 1 σ 1 + t 2 σ 2 + ⋯ + t β σ β ) = f ( t A ) = f ( 0 ) = 0 . Therefore, y ∈ r M β l S ( β ) ( A ) , i.e., y ∈ A M α by (5). It is easy to check that σ is an isomorphism.

(4) ⇒ (5) Clearly, A M α ⊆ r M β l S ( β ) ( A ) . Suppose A = σ 11 σ 12 ⋯ σ 1 α σ 21 σ 22 ⋯ σ 2 α ⋮ ⋮ ⋮ ⋮ σ β 1 σ β 2 ⋯ σ β α ∈ RFM β × α ( S ) and the t -th row of A is σ t = ( σ t 1 , σ t 2 , … , σ t α ) ∈ S ( α ) , t = 1 , 2 , … , β . Let y = y 1 y 2 ⋮ y β ∈ r M β l S ( β ) ( A ) . Define a left S -homomorphism g : S ( β ) A → M via g ( ( b 1 , b 2 , … , b β ) A ) = ( b 1 , b 2 , … , b β ) y , ∀ ( b 1 , b 2 , … , b β ) ∈ S ( β ) . It is easy to check that g is well defined. Thus, y = g ( σ 1 ) g ( σ 2 ) ⋮ g ( σ β ) . It follows that y ∈ A M α by (4). Hence, (5) holds.

(5) ⇒ (6) If 0 ≠ x ∈ M β ∕ A M α , then x ∉ A M α . By (5), we obtain that x ∉ r M β l S ( β ) ( A ) . Thus, there exists t ∈ l S ( β ) ( A ) such that t A = 0 and t x ≠ 0 . Define a right R -homomorphism θ : M β ∕ A M α → M via θ ( y ) = t y , for any y ∈ M β ∕ A M α . Then θ is well defined and θ ( x ) ≠ 0 . By [14, Corollary 8.13], we obtain that the right R -module M β ∕ A M α is cogenerated by M .

(6) ⇒ (5) Suppose A = σ 11 σ 12 ⋯ σ 1 α σ 21 σ 22 ⋯ σ 2 α ⋮ ⋮ ⋮ ⋮ σ β 1 σ β 2 ⋯ σ β α and y = y 1 y 2 ⋮ y β ∈ r M β l S ( β ) ( A ) . By the proof of ( 4 ) ⇔ ( 5 ) , there exists a left S -homomorphism f : S ( β ) A → M such that f ( ( σ i 1 , σ i 2 , … , σ i α ) ) = y i , i = 1 , 2 , … , β . Let g : M β ∕ A M α → M be any right R -homomorphism, π : M β → M β ∕ A M α the canonical epimorphism and η i : M → M β the i th injection. Hence,

0 = g π A x 1 x 2 ⋮ x α = g π σ 11 ( x 1 ) + σ 12 ( x 2 ) + ⋯ + σ 1 α ( x α ) σ 21 ( x 1 ) + σ 22 ( x 2 ) + ⋯ + σ 2 α ( x α ) ⋮ σ β 1 ( x 1 ) + σ β 2 ( x 2 ) + ⋯ + σ β α ( x α ) = g π ∑ i = 1 β ( η i σ i 1 ( x 1 ) + η i σ i 2 ( x 2 ) + ⋯ + η i σ i α ( x α ) ) = ∑ i = 1 β ( g π η i σ i 1 ( x 1 ) + g π η i σ i 2 ( x 2 ) + ⋯ + g π η i σ i α ( x α ) ) = ∑ i = 1 β g π η i ( σ i 1 , σ i 2 , … , σ i α ) x 1 x 2 ⋮ x α ,

for any x 1 x 2 ⋮ x α ∈ M α . Since g π η i ∈ S = End ( M R ) ,

g π ( y ) = g π ∑ i = 1 β η i f ( ( σ i 1 , σ i 2 , … , σ i α ) ) = ∑ i = 1 β g π η i f ( ( σ i 1 , σ i 2 , … , σ i α ) ) = ∑ i = 1 β f ( g π η i ( σ i 1 , σ i 2 , … , σ i α ) ) = f ∑ i = 1 β ( g π η i ( σ i 1 , σ i 2 , … , σ i α ) ) = f ( g π A ) = 0 .

If π ( y ) ≠ 0 , there exists right R -homomorphism g : M β ∕ A M α → M such that g π ( y ) ≠ 0 by (6), a contradiction. This implies that π ( y ) = 0 . Therefore, y ∈ A M α .□

Given a right R -module M , it is well known that there is an evaluation map σ M : M → Hom R ( Hom R ( M , R ) , R ) such that ( f ) σ M ( x ) = f ( x ) ( ∀ x ∈ M , f ∈ Hom R ( M , R ) ). In addition, M is said to be torsionless if σ M is a monomorphism.

Set M = R R in Theorem 2.2, we have the following.

Corollary 2.3

Let R be a ring and A ∈ R F M β × α ( R ) , the following conditions are equivalent.

  1. R R is injective relative to A .

  2. E x t R 1 ( R ( α ) ∕ R ( β ) A , R ) = 0 .

  3. σ : R α ∕ r R α ( A ) → Hom R ( R ( β ) A , R ) defined via

    σ ( x + r R α ( A ) ) ( b A ) = b A x , ( ∀ x ∈ R α , b ∈ R ( β ) )

    is an isomorphism of abelian groups.

  4. σ : A R α → Hom R ( R ( β ) A , R ) defined via

    σ ( A x ) ( b A ) = b A x , ( ∀ x ∈ R α , b ∈ R ( β ) )

    is an isomorphism of abelian groups.

  5. r R β l R ( β ) ( A ) = A R α .

  6. The right R -module R β ∕ A R α is torsionless.

Recall that S is called left injective relative to A [9] in case that for every h ∈ Hom S ( S ( β ) A , S ) , there exists g ∈ Hom S ( S ( α ) , S ) such that h = g η , i.e., the following diagram commutes

where η : S ( β ) S A → S S ( α ) is the canonical inclusion.

Corollary 2.4

[9] Let M be a right R-module with S = E n d ( M R ) and A ∈ R F M β × α ( S ) , the following conditions are equivalent.

  1. S S is injective relative to A .

  2. E x t S 1 ( S ( α ) ∕ S ( β ) A , S ) = 0 .

  3. r S β l S ( β ) ( A ) = A S α .

Definition 2.5

Let M be a right R -module with S = End ( M R ) . M is called endo ( α , β ) -injective if it is injective relative to all row-finite matrices A ∈ RFM β × α ( S ) .

Remark 2.6

Let M be a right R -module with S = End ( M R ) .

  1. M is endo ( m , n ) -injective [3] if it is left injective relative to all A ∈ S n × m .

  2. M is endoFP-injective [1] if it is endo ( m , n ) -injective, for all positive integers m , n .

  3. M is endoinjective [1] if and only if S M is endo ( α , β ) -injective for any cardinal numbers α and β . According to Baer’s famous criterion for injective modules, it is easy to see that M is endoinjective if and only if it is endo ( 1 , ∣ S ∣ ) -injective, where ∣ S ∣ is the cardinality of S .

Definition 2.7

Let M be a right R -module with S = End ( M R ) and A ∈ RFM β × α ( S ) .

  1. M is called quasi-projective relative to A in case that for every h ∈ Hom R ( M , A M α ) , there exists g ∈ Hom R ( M , M α ) such that h = π g , i.e., the following diagram commutes

    where π : M R α → A M R α is the canonical epimorphism.

  2. M is called endoquasi- ( β × α ) -projective if it is quasi-projective relative to all A ∈ RFM β × α ( S ) .

Lemma 2.8

Let M be a right R-module with S = E n d ( M R ) and A ∈ R F M β × α ( S ) . If S S is injective relative to A , then the following hold.

  1. If σ : M → M β is a right R-homomorphism with Im σ ⊆ A M α , then σ ∈ A S α .

  2. M is quasi-projective relative to A .

Proof

(1) Let a ∈ M . If σ = σ 1 σ 2 ⋮ σ β : M → M β , where Im σ ⊆ A M α , then there exists an element x = x 1 x 2 ⋮ x α ∈ M α such that

σ ( a ) = σ 1 σ 2 ⋮ σ β ( a ) = σ 1 ( a ) σ 2 ( a ) ⋮ σ β ( a ) = A x 1 x 2 ⋮ x α = A x .

If t ∈ l S ( β ) ( A ) , then t σ ( a ) = t A x = 0 . It follows that t σ = 0 , that is, σ ∈ r S β l S ( β ) ( A ) . Note that S S is injective relative to A . By Corollary 2.4, we obtain that σ ∈ A S α .

(2) Suppose σ = σ 1 σ 2 ⋮ σ β ∈ Hom R ( M , A M α ) . Then there is ϕ = ϕ 1 ϕ 2 ⋮ ϕ α ∈ S α such that σ = A ϕ by (1). This implies that σ = π ϕ , where π : M R α → A M R α is the canonical epimorphism.□

Lemma 2.9

Let M be a right R-module with S = E n d ( M R ) , A ∈ R F M β × α ( S ) and S M injective relative to A . If σ = σ 1 σ 2 ⋮ σ β ∈ r S β l S ( β ) ( A ) , then Im σ ⊆ A M α .

Proof

Let σ = σ 1 σ 2 ⋮ σ β ∈ r S β l S ( β ) ( A ) . Then t σ = ( t 1 , t 2 , … , t β ) σ 1 σ 2 ⋮ σ β = 0 , ∀ t = ( t 1 , t 2 , … , t β ) ∈ l S ( β ) ( A ) . This means that ( t 1 , t 2 , … , t β ) σ 1 σ 2 ⋮ σ β ( a ) = 0 , for any a ∈ M . It follows that σ 1 ( a ) σ 2 ( a ) ⋮ σ β ( a ) ∈ r M β l S ( β ) ( A ) . Note that S M is injective relative to A . Therefore, σ 1 ( a ) σ 2 ( a ) ⋮ σ β ( a ) ∈ A M α by Theorem 2.2, that is, Im σ ⊆ A M α .□

Theorem 2.10

Let M be a right R-module with S = E n d ( M R ) , A ∈ R F M β × α ( S ) and S M injective relative to A. The following conditions are equivalent.

  1. S is left injective relative to A .

  2. σ ∈ A S α , for any right R-morphism σ : M → M β with Im σ ⊆ A M α .

  3. M is quasi-projective relative to A.

Proof

(1) ⇒ (2) ⇒ (3) follows from Lemma 2.8.

(3) ⇒ (1) If σ = σ 1 σ 2 ⋮ σ β ∈ r S β l S ( β ) ( A ) , then Im σ ⊆ A M α by Lemma 2.9. This implies that σ ∈ Hom R ( M , A M α ) . By (3), there is a right R -morphism θ = θ 1 θ 2 ⋮ θ α : M → M α such that σ = π θ , where π : M R α → A M R α is the canonical epimorphism. Hence, σ ( a ) = π θ ( a ) = π θ 1 ( a ) θ 2 ( a ) ⋮ θ α ( a ) = A θ 1 ( a ) θ 2 ( a ) ⋮ θ α ( a ) = A θ 1 θ 2 ⋮ θ α ( a ) , for any a ∈ M . It follows that σ = A θ ∈ A S α . Thus, (1) holds.□

Remark 2.11

The condition “ S M is injective relative to A ” in Theorem 2.10 is necessary. In fact, let M = R R . Then M is quasi-projective relative to any matrix A . But S = End ( R R ) = R is not necessarily left injective relative to A .

Corollary 2.12

Let M be a right R-module with S = E n d ( M R ) and S M endo ( α , β ) -injective. The following conditions are equivalent.

  1. S is a left ( α , β ) -injective ring.

  2. M is endoquasi- ( α , β ) -projective.

Corollary 2.13

Let M be a right R-module with S = E n d ( M R ) and S M endo ( m , n ) -injective. The following conditions are equivalent.

  1. S is a left ( m , n ) -injective ring.

  2. M is endoquasi- ( m , n ) -projective.

Remark 2.14

From Theorem 2.10, Corollary 2.13 and [15, Proposition 4.33], we obtain a new necessary and sufficient condition of left endo ( m , n ) -injective rings in [3, Theorem 2.4].

Corollary 2.15

Let M be a right R-module with S = E n d ( M R ) and M is endoinjective. The following conditions are equivalent.

  1. S is a left injective ring.

  2. M is endoquasi- ( 1 , ∣ S ∣ ) -projective.

3 Coherence relative to matrices

Definition 3.1

Let M be a right R -module with S = End ( M R ) and A ∈ RFM β × α ( S ) . S S is called left coherent relative to A if Ker ( S ( β ) S → S ( β ) S A ) is finitely generated.

Recall that a left S -module S M is said to be S -Mittag Leffler ( S -ML) [16] in case the canonical map μ M , I : S I ⊗ M → M I defined via μ M , I ( ( s i ) ⊗ x ) = ( s i x ) ( ∀ x ∈ M and ( s i ) ∈ S I ) is a monomorphism for every set I . It is well known that S M is finitely presented if and only if M is finitely generated and S -ML.

Remark 3.2

A ring S is said to be left coherent relative to A ∈ S β × α [8,9] if S ( β ) A is a left S -ML module. Note that S ( β ) A is a β -generated submodule of S α in Definition 3.1. Hence, if β is finite, S ( β ) A is finitely presented if and only if S ( β ) A is S -ML. Therefore, the left coherent relative to A here coincides with the definition in [8, 9] when β is finite.

Definition 3.3

Let M be a right R -module with S = End ( M R ) . S is called left ( α , n ) -coherent if it is left coherent to all matrices A ∈ RFM n × α ( S ) .

Remark 3.4

  1. S is left ( m , n ) -coherent [17] if it is left coherent relative to all A ∈ S n × m .

  2. A ring S is a left coherent ring if and only if S is left ( m , n ) -coherent for all positive integer m , n . If A ∈ RFM n × α ( S ) , then there exists an integer l such that S n A ⊆ S l . Thus, a ring S is a left coherent ring if and only if S is left ( α , β ) -coherent for all β ∈ N and all cardinal numbers α .

Let C be a class of left R -modules and M a left R -module. Following [18], we say that a homomorphism φ : M → C is a C -preenvelope of M if C ∈ C and the abelian group homomorphism Hom ( φ , C ′ ) : Hom ( C , C ′ ) → Hom ( M , C ′ ) is surjective for each C ′ ∈ C . A C -preenvelope φ : M → C is called a C -envelope if every endomorphism f : C → C such that f φ = φ is an isomorphism. Dually, we have the definitions of C -precovers and C -covers. C -envelopes ( C -covers) may not exist in general, but if they exist, they are unique up to isomorphisms.

Theorem 3.5

Let M be a right R-module with S = E n d ( M R ) and A ∈ R F M n × α ( S ) . Then the following conditions are equivalent.

  1. S is left coherent relative to A .

  2. Hom R ( M n ∕ A M α , M ) is a finitely generated left S-module.

  3. M n ∕ A M α has an a d d ( M ) -preenvelope.

Proof

Consider the exact sequence

0 ⟶ A M α ⟶ θ M n ⟶ M n ∕ A M α ⟶ 0 ,

which deduces the exact sequence of left S -modules

0 ⟶ Hom R ( M n ∕ A M α , M ) ⟶ Hom R ( M n , M ) ⟶ θ ∗ Hom R ( A M α , M ) .

Thus, Hom R ( M n , M ) ≅ S ( n ) and im ( θ ∗ ) ≅ S ( n ) A .

( 1 ) ⇒ ( 2 ) By the definition, we obtain that Hom R ( M n ∕ A M α , M ) is finitely generated.

( 2 ) ⇒ ( 3 ) By (2), there is a generating set { g i ∈ Hom R ( M n ∕ A M α , M ) : 1 ≤ i ≤ k } of Hom R ( M n ∕ A M α , M ) . Define a right R -homomorphism g : M n ∕ A M α → M k via g ( a ) = g 1 ( a ) g 2 ( a ) ⋮ g k ( a ) , ∀ a ∈ M n ∕ A M α . Thus, g is an add ( M ) -preenvelope of M n ∕ A M α . In fact, let m be any positive integer and ψ : M n ∕ A M α → M m any right R -morphism. Thus, we set ψ = ψ 1 ψ 2 ⋮ ψ m via ψ ( a ) = ψ 1 ( a ) ψ 2 ( a ) ⋮ ψ m ( a ) ( ∀ a ∈ M n ∕ A M α ) , where ψ i = π i ψ and π i : M m → M is the projection. Then there exists h i j ∈ S ( 1 ≤ i ≤ m , 1 ≤ j ≤ k ) such that ψ i = ∑ j = 1 k h i j g j , i = 1 , 2 , … , m . Define a right R -morphism h = ( h i j ) i × j : M m → M k . It is easy to check that ψ = h g . So g is an add ( M ) -precover.

( 3 ) ⇒ ( 1 ) By (3), M n ∕ A M α has an add ( M ) -preenvelope M n ∕ A M α → M k . There exists an exact sequence of left S -homomorphisms

Hom R ( M k , M ) → Hom R ( M n ∕ A M α , M ) → 0 .

Note that Hom R ( M k , M ) ≅ S ( k ) . It follows that Hom R ( M n ∕ A M α , M ) is a finitely generated left S -module. Hence, S is a left coherent ring relative to A .□

Corollary 3.6

Let M be a right R-module with S = E n d ( M R ) and A ∈ R F M n × α ( S ) . Then the following conditions are equivalent.

  1. S is left coherent relative to A and S M is injective relative to A .

  2. M n ∕ A M α has a monic a d d ( M ) -preenvelope.

Proof

( 1 ) ⇒ ( 2 ) Since S M is injective relative to A , there exists a right R -monomorphism g : M n ∕ A M α → M I by Theorem 2.2. Let p i : M I → M be the canonical projection. Since S is left coherent relative to A , M n ∕ A M α has an add ( M ) -preenvelope f : M n ∕ A M α → M k by Theorem 3.5. Hence, there exists θ i : M k → M such that p i g = θ i f , for any i ∈ I . Since g is monic, g ( x ) ≠ 0 , for any 0 ≠ x ∈ M n ∕ A M α . Therefore, there exists i ∈ I such that p i g ( x ) ≠ 0 . It follows that f ( x ) ≠ 0 , that is, f is a monic add ( M ) -preenvelope.

( 2 ) ⇒ ( 1 ) follows by Theorems 3.5 and 2.2.□

In a similar way to [19, Proposition 1] and [2, Theorem 2.7], we obtain the following Proposition, which is parallel with [2, Theorem 2.7]. And, we omit the proof.

Proposition 3.7

Let M be a right R-module with S = E n d ( M R ) and A ∈ R F M n × α ( S ) . Then the following conditions are equivalent.

  1. S is left coherent relative to A and Hom R ( M n ∕ A M α , M ) (as a left S-module) has a projective cover g : P → Hom R ( M n ∕ A M α , M ) ( P is a projective left S-module).

  2. M n ∕ A M α has an a d d ( M ) -envelope f : M n ∕ A M α → F .

Moreover,

f ∗ : Hom R ( M n ∕ A M α , M ) → Hom R ( F , M ) is a projective cover of Hom R ( M n ∕ A M α , M ) .

g ∗ β M n ∕ A M α is an a d d ( M ) -envelope of M n ∕ A M α , where g ∗ : Hom S ( Hom R ( M n ∕ A M α , M ) , M ) → Hom S ( P , M ) and β M n ∕ A M α : M n ∕ A M α → Hom S ( Hom R ( M n ∕ A M α , M ) , M ) is the evaluation morphism.

Recall that a C -envelope φ : X → C of X is said to have the unique mapping property [20] if for any homomorphism, f : X → C ′ with C ′ ∈ C , there is a unique homomorphism g : C → C ′ such that f = g φ .

Proposition 3.8

Let M be a right R-module with S = E n d ( M R ) and A ∈ R F M n × α ( S ) . Then the followings are equivalent.

  1. Hom R ( M n ∕ A M α , M ) is a finitely generated projective left S -module.

  2. M n ∕ A M α has an a d d ( M ) -envelope with the unique mapping property.

Proof

( 2 ) ⇒ ( 1 ) In view of (1), M n ∕ A M α has an add ( M ) -envelope with the unique mapping property f : M n ∕ A M α → H . Consider the following exact sequence of right R -modules

M n ∕ A M α → H → c o k e r ( f ) → 0 ,

which induces the exact sequence of left S -modules

0 → Hom R ( c o k e r ( f ) , M ) → Hom R ( H , M ) → Hom R ( M n ∕ A M α , M ) → 0 .

Note that f has the unique mapping property. Then Hom R ( c o k e r ( f ) , M ) = 0 . It follows that Hom R ( H , M ) ≅ Hom R ( M n ∕ A M α , M ) . Thus, Hom R ( M n ∕ A M α , M ) is a finitely generated projective left S -module because H is a direct summand of M k , for some integer k .

( 1 ) ⇒ ( 2 ) Since Hom R ( M n ∕ A M α , M ) is finitely generated projective, the evaluation morphism

β M n ∕ A M α : M n ∕ A M α → Hom S ( Hom R ( M n ∕ A M α , M ) , M )

is an add ( M ) -envelope of M n ∕ A M α by Corollary 3.7. Thus,

β ∗ : Hom R ( Hom S ( Hom R ( M n ∕ A M α , M ) , M ) , M ) → Hom R ( M n ∕ A M α , M )

is an isomorphism because Hom R ( M n ∕ A M α , M ) is a finitely generated projective left S -module. It follows that β M n ∕ A M α is an add ( M ) -envelope with the unique mapping property.□

Theorem 3.9

Let M be a right R-module with S = E n d ( M R ) and A ∈ R F M n × α ( S ) . Then the following are equivalent:

  1. S is left coherent relative to A , and any right R -homomorphism f : M n ∕ A M α → K 1 factors through a right R-module in a d d ( M ) , where K 1 is any submodule of a right R-module K ∈ a d d ( M ) .

  2. M n ∕ A M α has an epic a d d ( M ) -envelope.

Proof

( 1 ) ⇒ ( 2 ) S is left coherent relative to A by Theorem 3.5. By (2), M n ∕ A M α has an epic add ( M ) -envelope g : M n ∕ A M α → P ( P ∈ add ( M ) ) . Let K 1 be a submodule of any right R -module K ∈ add ( M ) . For any f ∈ Hom ( M n ∕ A M α , K 1 ) , there is a right R -homomorphism ψ : P → L such that ψ g = i f , where i : K 1 → K is the inclusion map. Consider the following diagram:

Define s : P → K 1 via s ( g ( x ) ) = f ( x ) for x ∈ M n ∕ A M α . It is clear that s is well-defined and f = s g . In fact, if g ( x ) = 0 , then 0 = ψ g ( x ) = i f ( x ) . It follows that f ( x ) = 0 since i is a monomorphism. So, f factors through a right R -module P ∈ add ( M ) .

( 2 ) ⇒ ( 1 ) Since S is left coherent relative to A , M n ∕ A M α has an add ( M ) -preenvelope f : M n ∕ A M α → F by Theorem 3.5. Let F 1 = i m ( f ) , θ : M n ∕ A M α → F 1 the induced epimorphism of f and i : F 1 → F the inclusion. By (1), there exist a right R -module H ∈ add ( M ) and right R -morphisms g : M n ∕ A M α → H and h : H → F 1 such that θ = h g . Note that f is a preenvelope. There exists φ : F → H such that g = φ f . Thus, θ = h g = h φ f = h φ i θ , and so I d F 1 = h φ i since θ is epic. Hence, F 1 ∈ add ( M ) . It is easy to see that θ is an epic add ( M ) -envelope of M n ∕ A M α .□

Corollary 3.10

Let M be a right R-module with S = E n d ( M R ) and A ∈ R F M n × α ( S ) . Then the following are equivalent:

  1. M n ∕ A M α ∈ a d d ( M ) .

  2. M n ∕ A M α has an epic a d d ( M ) -envelope and S M is injective relative to A .

Proof

( 1 ) ⇒ ( 2 ) is trivial.

( 2 ) ⇒ ( 1 ) By ( 2 ) , we obtain that M n ∕ A M α has an epic add ( M ) -envelope f : M n ∕ A M α → H ( H ∈ add ( M ) ) . Since S M is injective relative to A , f is monic by Corollary 3.6. Thus, f is an isomorphism and (1) holds.□

Definition 3.11

A left R -module L is called M - ( α , β ) -presented if there exists an exact sequence N ( β ) → N ( α ) → L → 0 of left R -modules.

Corollary 3.12

Let M be a right R-module with S = E n d ( M R ) .

  1. [2, Theorem 2.1] S is left ( m , n ) -coherent if and only if every M - ( n , m ) -presented right R-module has an a d d ( M ) -preenvelope.

  2. [2, Theorem 2.11] S is left ( m , n ) -coherent and S M is endo ( m , n ) -injective if and only if every M - ( n , m ) -presented right R-module has a monic a d d ( M ) -preenvelope.

  3. [2, Theorem 2.7] S is left ( m , n ) -coherent and Hom R ( L , M ) has a projective cover for every M - ( n , m ) -presented right R-module L if and only if every M- ( n , m ) -presented right R -module has an a d d ( M ) -envelope.

  4. [2, Theorem 2.8] Hom R ( L , M ) is a finitely generated projective left S-module for every M - ( n , m ) -presented right R-module L if and only if every M - ( n , m ) -presented right R -module has an a d d ( M ) -envelope with the unique mapping property.

  5. S is left ( m , n ) -coherent and any right R -homomorphism f : L → K 1 factors through a module in a d d ( N ) , where L is any M - ( n , m ) -presented right R-module and K 1 is any submodule of a left R -module K ∈ a d d ( N ) if and only if every M - ( n , m ) -presented right R-module has an epic a d d ( M ) -envelope.

  6. Every M - ( n , m ) -presented right R-module belongs to a d d ( M ) if and only if every M - ( n , m ) -presented right R-module has an epic a d d ( M ) -envelope and S M is endo ( m , n ) -injective relative to A .

Remark 3.13

Corollary 3.12 (5) gives a new characterization of [2, Theorem 2.9].

Theorem 3.14

Let M be a right R-module with S = E n d ( M R ) and A ∈ R F M n × α ( S ) . The following are equivalent.

  1. S ( n ) A is a projective left S-module and S M is injective relative to A .

  2. M n ∕ A M α is a direct summand of M n .

Proof

Since A ∈ RFM n × α ( S ) , there is an integer l such that A = ( A 1 , 0 ) , where A 1 is an n × l matrix. Thus, A M α = A 1 M l . Let L = M n ∕ A M α = M n ∕ A 1 M l . Then there is an exact sequence

0 ⟶ A 1 M l ⟶ i M n ⟶ φ L ⟶ 0 ,

which deduces the following diagram of left S -modules

where Hom R ( M n , M ) ≅ S ( n ) and im ( i ∗ ) ≅ S ( n ) A 1 .

( 1 ) ⇒ ( 2 ) Since S ( n ) A is projective, S ( n ) A 1 ≅ S ( n ) A = S ( n ) ( A 1 , 0 ) is projective. It means that φ ∗ is a spilt monomorphism by the aforementioned diagram. Thus, φ ∗ ∗ is a split epimorphism. Since S M is injective relative to A , it is easy to check that S M is injective relative to A 1 . Hence, we have the following exact sequence by Definition 2.1:

Hom S ( S ( l ) , M ) ⟶ Hom S ( S ( n ) , M ) ⟶ φ ∗ ∗ Hom S ( Hom R ( L , M ) , M ) ⟶ 0 .

Consider the following exact sequence

By the five lemma, β L is an isomorphism. Note that φ ∗ ∗ is a spilt epimorphism. Then φ is split by the aforementioned diagram. Hence L is a direct summand of M n .

( 2 ) ⇒ ( 1 ) Since L is a direct summand of M n , φ ∗ is split by diagram 1 ○ . It follows that S ( n ) A ≅ S ( n ) A 1 is a projective left S -module. According to (2), we obtain that L is finitely cogenerated by M . Thus, S M is injective relative to A by Theorem 2.2.□

Recall that a ring R is called left n-semihereditary [21,22] if every n -generated right ideal of R is projective, equivalently, if every n -generated submodule of a projective right R -module is projective.

Corollary 3.15

Let M be a right R -module with S = E n d ( M R ) . The followings are equivalent.

  1. S is a left n-semihereditary ring and S M is end ( m , n ) -injective.

  2. every M - ( n , m ) -presented right R-module is a direct summand of M n .

Proof

It is trivial by Remark 3.4 (1) and Theorem 3.14.□

Let M be a right R -module with S = End ( M R ) . Following [4], M is called endoregular if S is a von Neumann regular ring. Lee et al. provided characterizations of endoregular modules in [4]. Moreover, it is well known that S is a von Neumann regular ring if and only if every left(right) S -module is (1,1)-injective if and only if every left(right) S -module is ( m , n ) -injective if and only if every left(right) S -module is ( 1 , 1 ) -flat if and only if every left(right) S -module is ( m , n ) -flat (see [7, Corollary 2.6], [23, Corollary 2.6], [24, Theorem 4.1.1(b)]).

By [4, Corollary 3.15], we obtain that if S is a von Neumann regular ring, then M R is endoquasi- ( m , n ) -projective. Thus, we obtain the following Corollary.

Corollary 3.16

Let M be a right R -module with S = E n d ( M R ) . The followings are equivalent:

  1. S is a left n-semihereditary ring and S M is endo ( m , n ) -injective and endoquasi- ( m , n ) -projective.

  2. S is a von Neumann regular ring.

Proof

( 1 ) ⇒ ( 2 ) Since M is endoquasi- ( m , n ) -projective and endo ( m , n ) -injective, S is a left ( m , n ) -injective ring by Corollary 2.13, and so is S ( I ) , for any set I . By [22, Theorem 3] and (1), each quotient-module of an ( m , n ) -injective left S -module is ( m , n ) -injective. It follows that every left S -module is ( m , n ) -injective. Thus, S is a von Neumann regular ring.

( 2 ) ⇒ ( 1 ) It is trivial.□

Let M R = R R .

Corollary 3.17

The following are equivalent for a ring R .

  1. R is left n-semihereditary and left ( m , n ) -injective.

  2. R is a von Neumann regular ring.

4 Conclusions

We conclude this article with the following.

Remark 4.1

It would be interesting to extend the results to coherent [13], n -coherent [25] and P -coherent [26] endomorphism rings. For example, a right R -module L is called finitely M-presented [1] if there exist positive integers m , n and an exact sequence M m → M n → L → 0 of right R -modules. Let W D ( S ) denote the weak global dimension of S . Then S is left coherent if and only if every finitely M -presented module has an add ( M ) -preenvelope by Theorem 3.5; S is left coherent and W D ( S ) ≤ 2 if and only if every finitely M -presented module has an add ( M ) -envelope with the unique mapping property if and only if Hom R ( L , M ) is a finitely generated projective left S -module ( L is any finitely M -presented module) by [1, Theorem 2.9] and Proposition 3.8.

Remark 4.2

The computation of the endomorphism rings is an important problem in computational number theory as well as in cryptography. For instance, it is important in the computation of class polynomials, which play an important role in explicit class field theory. There are many calculations and much research on endomorphism rings, such as [27–29], etc.

Remark 4.3

Let N be a left R -module with the endomorphism ring S = End ( N R ) . Duality to injective modules relative to a matrix and add ( N ) -envelopes, flat modules relative to a matrix and add ( N ) -covers may also be studied similarly.

Acknowledgements

Thanks for the referees’ comments. Those comments are all valuable and very helpful for revising and improving our paper, as well as the important guiding significance to our researches. The work was supported by Natural Science Foundation of Fujian Province, China (2020J01908), Educational Commission of Fujian Province, China (JAT190582).

  1. Conflict of interest: The author declares that he has no conflicts of interest.

References

[1] J. L. Garcia, J. Martinez Hernandez, and P. L. Gomez Sanchez, Endomorphism rings and the category of finitely generated projective modules, Comm. Algebra 28 (2000), no. 8, 3837–3852, DOI: http://doi.org/10.1080/00927870008827061. 10.1080/00927870008827061Search in Google Scholar

[2] X. Q. Luo and L. X. Mao, Modules whose endomorphism rings are (m,n)-coherent, Algebra Colloq. 26 (2019), no. 2, 231–242, DOI: http://doi.org/10.1142/S100538671900018X. 10.1142/S100538671900018XSearch in Google Scholar

[3] L. X. Mao, (m,n)-injective and (m,n)-coherent endomorphism rings of modules, J. Algebra Appl. 19 (2020), no. 3, 2050048. 10.1142/S0219498820500486Search in Google Scholar

[4] G. Lee, S. T. Rizvi, and C. Roman, Modules whose endomorphism rings are von Neumann regular, Comm. Algebra 41 (2013), no. 11, 4066–4088, DOI: http://dx.doi.org/10.1080/00927872.2012.700979. 10.1080/00927872.2012.700979Search in Google Scholar

[5] L. Angeleri-Hügel, Endocoherent modules, Pacific J. Math. 212 (2003), no. 1, 1–11, DOI: http://doi.org/10.2140/pjm.2003.212.1. 10.2140/pjm.2003.212.1Search in Google Scholar

[6] G. Azumaya, Finite splitness and finite projectivity, J. Algebra 106 (1987), no. 1, 114–134, DOI: http://dx.doi.org/10.1016/0021-8693(87)90024-X. 10.1016/0021-8693(87)90024-XSearch in Google Scholar

[7] N. Q. Ding and J. L. Chen, Relative coherence and preenvelopes, Manuscripta Math. 81 (1993), no. 3–4, 243–262, DOI: http://doi.org/10.1007/BF02567857. 10.1007/BF02567857Search in Google Scholar

[8] X. X. Zhang and J. L. Chen, Homological properties of modules characterized by matrices, J. Southeast Univ. (English Ed.) 21 (2005), no. 2, 239–243. Search in Google Scholar

[9] X. X. Zhang, J. L. Chen and J. Zang, Properties of modules and rings relative to some matrices, Comm. Algebra 36 (2008), no. 10, 3682–3707, DOI: http://dx.doi.org/10.1080/00927870802157996. 10.1080/00927870802157996Search in Google Scholar

[10] P. Jampachon, J. Ittharat, and N. V. Sanh, On finite injectivity, Southeast Asian Bull. Math. 24 (2000), no. 4, 559–564. Search in Google Scholar

[11] W. K. Nicholson and M. F. Yousif, Principally injective rings, J. Algebra 174 (1995), no. 1, 77–93, DOI: http://doi.org/10.1006/jabr.1995.1117. 10.1006/jabr.1995.1117Search in Google Scholar

[12] N. V. Sanh and K. P. Shum, Endomorphism rings of quasi-principally injective modules, Comm. Algebra 29 (2001), no. 4, 1437–1443, DOI: http://doi.org/10.1081/AGB-100002110. 10.1081/AGB-100002110Search in Google Scholar

[13] S. U. Chase, Direct products of modules, Trans. Amer. Math. Soc. 97 (1960), 457–473, DOI: http://doi.org/10.2307/1993382. 10.1090/S0002-9947-1960-0120260-3Search in Google Scholar

[14] F. W. Anderson and K. R. Fuller, Rings and Categries of Modules, Springer-Verlag, New York, 1974. 10.1007/978-1-4684-9913-1Search in Google Scholar

[15] S. H. Mohamed and B. J. Müller, Continuous and Discrect Modules, Cambridge University Press, Cambridge, 1990. 10.1017/CBO9780511600692Search in Google Scholar

[16] T. G. Clarke, On ℵ-Projective Modules, Ph.D. thesis, Kent State University, 1976. Search in Google Scholar

[17] X. X. Zhang, J. L. Chen, and J. Zang, On (m,n)-injective modules and (m,n)-coherent rings, Algebra Colloq. 12 (2005), 149–160. 10.1142/S1005386705000143Search in Google Scholar

[18] E. E. Enochs, Injective and flat covers, envelopes and resolvents, Israel J. Math. 39 (1981), 189–209, DOI: http://doi.org/10.1007/BF02760849. 10.1007/BF02760849Search in Google Scholar

[19] J. A. Mayor and J. M. Hernaaandez, Flat envelopes in commutative rings, Israel. J. Math. 62 (1988), no. 1, 123–128, DOI: http://doi.org/10.1007/BF02767358. 10.1007/BF02767358Search in Google Scholar

[20] N. Q. Ding, On envelopes with the unique mapping property, Comm. Algebra 24 (1996), no. 4, 1459–1470, DOI: http://dx.doi.org/10.1080/00927879608825646. 10.1080/00927879608825646Search in Google Scholar

[21] B. Stenström, Coherent rings and FP-injective modules, J. London Math. Soc. s2-2 (1970), no. 2, 323–329, DOI: http://dx.doi.org/10.1112/jlms/s2-2.2.323. 10.1112/jlms/s2-2.2.323Search in Google Scholar

[22] Z. M. Zhu and Z. S. Tan, On n-semihereditary rings, Sci. Math. Jpn. 62 (2005), no. 3, 455–459. Search in Google Scholar

[23] J. Dauns and L. Fuchs, Torsion-freeness for rings with zero-divisors, J. Algebra Appl. 3 (2004), no. 3, 221–237, DOI: http://dx.doi.org/10.1142/S0219498804000885. 10.1142/S0219498804000885Search in Google Scholar

[24] R. Göbel and J. Trlifaj, Approximations and Endomorphism Algebras of Modules, De Gruyter Expositions in Mathematics, Vol. 41, Walter de Gruyter, Berlin, 2006. 10.1515/9783110199727Search in Google Scholar

[25] A. Shamsuddin, n-injective and n-flat modules, Comm. Algebra 29 (2001), no. 5, 2039–2050, DOI: http://doi.org/10.1081/AGB-100002166. 10.1081/AGB-100002166Search in Google Scholar

[26] J. E. Björk, Rings satisfying certain chain conditions, J. Reine Angew. Math. 245 (1970), 63–73. 10.1515/crll.1970.245.63Search in Google Scholar

[27] A. Abyzov, R. Barati, and P. V. Danchev, Rings Close to Periodic with Applications to Matrix, Endomorphism and Group Rings, arXiv:2301.07948, 2023, https://doi.org/10.48550/arXiv.2301.07948. Search in Google Scholar

[28] J. F. Carlson, Negative cohomology and the endomorphism ring of the trivial module, J. Pure Appl. Algebra 226 (2022), no. 9, 107046, DOI: https://doi.org/10.1016/j.jpaa.2022.107046. 10.1016/j.jpaa.2022.107046Search in Google Scholar

[29] V. Fernandes and T. Paulista, On the rank of monoids of endomorphisms of a finite directed path, Asian-Eur J. Math. 16 (2023), no. 4, 2350069, DOI: https://doi.org/10.1142/S1793557123500699. 10.1142/S1793557123500699Search in Google Scholar
Received: 2023-02-13
Revised: 2023-06-03
Accepted: 2023-07-08
Published Online: 2023-09-04

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Special Issue on Future Directions of Further Developments in Mathematics
  2. What will the mathematics of tomorrow look like?
  3. On H 2-solutions for a Camassa-Holm type equation
  4. Classical solutions to Cauchy problems for parabolic–elliptic systems of Keller-Segel type
  5. Control of multi-agent systems: Results, open problems, and applications
  6. Logical perspectives on the foundations of probability
  7. Subharmonic solutions for a class of predator-prey models with degenerate weights in periodic environments
  8. A non-smooth Brezis-Oswald uniqueness result
  9. Luenberger compensator theory for heat-Kelvin-Voigt-damped-structure interaction models with interface/boundary feedback controls
  10. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part II)
  11. Positive solution for a nonlocal problem with strong singular nonlinearity
  12. Analysis of solutions for the fractional differential equation with Hadamard-type
  13. Hilfer proportional nonlocal fractional integro-multipoint boundary value problems
  14. A comprehensive review on fractional-order optimal control problem and its solution
  15. The θ-derivative as unifying framework of a class of derivatives
  16. Review Articles
  17. On the use of L-functionals in regression models
  18. Minimal-time problems for linear control systems on homogeneous spaces of low-dimensional solvable nonnilpotent Lie groups
  19. Regular Articles
  20. Existence and multiplicity of solutions for a new p(x)-Kirchhoff problem with variable exponents
  21. An extension of the Hermite-Hadamard inequality for a power of a convex function
  22. Existence and multiplicity of solutions for a fourth-order differential system with instantaneous and non-instantaneous impulses
  23. Relay fusion frames in Banach spaces
  24. Refined ratio monotonicity of the coordinator polynomials of the root lattice of type Bn
  25. On the uniqueness of limit cycles for generalized Liénard systems
  26. A derivative-Hilbert operator acting on Dirichlet spaces
  27. Scheduling equal-length jobs with arbitrary sizes on uniform parallel batch machines
  28. Solutions to a modified gauged Schrödinger equation with Choquard type nonlinearity
  29. A symbolic approach to multiple Hurwitz zeta values at non-positive integers
  30. Some results on the value distribution of differential polynomials
  31. Lucas non-Wieferich primes in arithmetic progressions and the abc conjecture
  32. Scattering properties of Sturm-Liouville equations with sign-alternating weight and transmission condition at turning point
  33. Some results for a p(x)-Kirchhoff type variation-inequality problems in non-divergence form
  34. Homotopy cartesian squares in extriangulated categories
  35. A unified perspective on some autocorrelation measures in different fields: A note
  36. Total Roman domination on the digraphs
  37. Well-posedness for bilevel vector equilibrium problems with variable domination structures
  38. Binet's second formula, Hermite's generalization, and two related identities
  39. Non-solid cone b-metric spaces over Banach algebras and fixed point results of contractions with vector-valued coefficients
  40. Multidimensional sampling-Kantorovich operators in BV-spaces
  41. A self-adaptive inertial extragradient method for a class of split pseudomonotone variational inequality problems
  42. Convergence properties for coordinatewise asymptotically negatively associated random vectors in Hilbert space
  43. Relating the super domination and 2-domination numbers in cactus graphs
  44. Compatibility of the method of brackets with classical integration rules
  45. On the inverse Collatz-Sinogowitz irregularity problem
  46. Positive solutions for boundary value problems of a class of second-order differential equation system
  47. Global analysis and control for a vector-borne epidemic model with multi-edge infection on complex networks
  48. Nonexistence of global solutions to Klein-Gordon equations with variable coefficients power-type nonlinearities
  49. On 2r-ideals in commutative rings with zero-divisors
  50. A comparison of some confidence intervals for a binomial proportion based on a shrinkage estimator
  51. The construction of nuclei for normal constituents of Bπ-characters
  52. Weak solution of non-Newtonian polytropic variational inequality in fresh agricultural product supply chain problem
  53. Mean square exponential stability of stochastic function differential equations in the G-framework
  54. Commutators of Hardy-Littlewood operators on p-adic function spaces with variable exponents
  55. Solitons for the coupled matrix nonlinear Schrödinger-type equations and the related Schrödinger flow
  56. The dual index and dual core generalized inverse
  57. Study on Birkhoff orthogonality and symmetry of matrix operators
  58. Uniqueness theorems of the Hahn difference operator of entire function with a Picard exceptional value
  59. Estimates for certain class of rough generalized Marcinkiewicz functions along submanifolds
  60. On semigroups of transformations that preserve a double direction equivalence
  61. Positive solutions for discrete Minkowski curvature systems of the Lane-Emden type
  62. A multigrid discretization scheme based on the shifted inverse iteration for the Steklov eigenvalue problem in inverse scattering
  63. Existence and nonexistence of solutions for elliptic problems with multiple critical exponents
  64. Interpolation inequalities in generalized Orlicz-Sobolev spaces and applications
  65. General Randić indices of a graph and its line graph
  66. On functional reproducing kernels
  67. On the Waring-Goldbach problem for two squares and four cubes
  68. Singular moduli of rth Roots of modular functions
  69. Classification of self-adjoint domains of odd-order differential operators with matrix theory
  70. On the convergence, stability and data dependence results of the JK iteration process in Banach spaces
  71. Hardy spaces associated with some anisotropic mixed-norm Herz spaces and their applications
  72. Remarks on hyponormal Toeplitz operators with nonharmonic symbols
  73. Complete decomposition of the generalized quaternion groups
  74. Injective and coherent endomorphism rings relative to some matrices
  75. Finite spectrum of fourth-order boundary value problems with boundary and transmission conditions dependent on the spectral parameter
  76. Continued fractions related to a group of linear fractional transformations
  77. Multiplicity of solutions for a class of critical Schrödinger-Poisson systems on the Heisenberg group
  78. Approximate controllability for a stochastic elastic system with structural damping and infinite delay
  79. On extremal cacti with respect to the first degree-based entropy
  80. Compression with wildcards: All exact or all minimal hitting sets
  81. Existence and multiplicity of solutions for a class of p-Kirchhoff-type equation RN
  82. Geometric classifications of k-almost Ricci solitons admitting paracontact metrices
  83. Positive periodic solutions for discrete time-delay hematopoiesis model with impulses
  84. On Hermite-Hadamard-type inequalities for systems of partial differential inequalities in the plane
  85. Existence of solutions for semilinear retarded equations with non-instantaneous impulses, non-local conditions, and infinite delay
  86. On the quadratic residues and their distribution properties
  87. On average theta functions of certain quadratic forms as sums of Eisenstein series
  88. Connected component of positive solutions for one-dimensional p-Laplacian problem with a singular weight
  89. Some identities of degenerate harmonic and degenerate hyperharmonic numbers arising from umbral calculus
  90. Mean ergodic theorems for a sequence of nonexpansive mappings in complete CAT(0) spaces and its applications
  91. On some spaces via topological ideals
  92. Linear maps preserving equivalence or asymptotic equivalence on Banach space
  93. Well-posedness and stability analysis for Timoshenko beam system with Coleman-Gurtin's and Gurtin-Pipkin's thermal laws
  94. On a class of stochastic differential equations driven by the generalized stochastic mixed variational inequalities
  95. Entire solutions of two certain Fermat-type ordinary differential equations
  96. Generalized Lie n-derivations on arbitrary triangular algebras
  97. Markov decision processes approximation with coupled dynamics via Markov deterministic control systems
  98. Notes on pseudodifferential operators commutators and Lipschitz functions
  99. On Graham partitions twisted by the Legendre symbol
  100. Strong limit of processes constructed from a renewal process
  101. Construction of analytical solutions to systems of two stochastic differential equations
  102. Two-distance vertex-distinguishing index of sparse graphs
  103. Regularity and abundance on semigroups of partial transformations with invariant set
  104. Liouville theorems for Kirchhoff-type parabolic equations and system on the Heisenberg group
  105. Spin(8,C)-Higgs pairs over a compact Riemann surface
  106. Properties of locally semi-compact Ir-topological groups
  107. Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2
  108. Ordering stability of Nash equilibria for a class of differential games
  109. A new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order
  110. About a dubious proof of a correct result about closed Newton Cotes error formulas
  111. Ricci ϕ-invariance on almost cosymplectic three-manifolds
  112. Schur-power convexity of integral mean for convex functions on the coordinates
  113. A characterization of a ∼ admissible congruence on a weakly type B semigroup
  114. On Bohr's inequality for special subclasses of stable starlike harmonic mappings
  115. Properties of meromorphic solutions of first-order differential-difference equations
  116. A double-phase eigenvalue problem with large exponents
  117. On the number of perfect matchings in random polygonal chains
  118. Evolutoids and pedaloids of frontals on timelike surfaces
  119. A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine
  120. The 𝔪-WG° inverse in the Minkowski space
  121. Stability result for Lord Shulman swelling porous thermo-elastic soils with distributed delay term
  122. Approximate solvability method for nonlocal impulsive evolution equation
  123. Construction of a functional by a given second-order Ito stochastic equation
  124. Global well-posedness of initial-boundary value problem of fifth-order KdV equation posed on finite interval
  125. On pomonoid of partial transformations of a poset
  126. New fractional integral inequalities via Euler's beta function
  127. An efficient Legendre-Galerkin approximation for the fourth-order equation with singular potential and SSP boundary condition
  128. Eigenfunctions in Finsler Gaussian solitons
  129. On a blow-up criterion for solution of 3D fractional Navier-Stokes-Coriolis equations in Lei-Lin-Gevrey spaces
  130. Some estimates for commutators of sharp maximal function on the p-adic Lebesgue spaces
  131. A preconditioned iterative method for coupled fractional partial differential equation in European option pricing
  132. A digital Jordan surface theorem with respect to a graph connectedness
  133. A quasi-boundary value regularization method for the spherically symmetric backward heat conduction problem
  134. The structure fault tolerance of burnt pancake networks
  135. Average value of the divisor class numbers of real cubic function fields
  136. Uniqueness of exponential polynomials
  137. An application of Hayashi's inequality in numerical integration
https://doi.org/10.1515/math-2023-0612
Keywords for this article
coherent; injective; endomorphism ring; preenvelope; quasi-projective
Creative Commons
BY 4.0

Articles in the same Issue

  1. Special Issue on Future Directions of Further Developments in Mathematics
  2. What will the mathematics of tomorrow look like?
  3. On H 2-solutions for a Camassa-Holm type equation
  4. Classical solutions to Cauchy problems for parabolic–elliptic systems of Keller-Segel type
  5. Control of multi-agent systems: Results, open problems, and applications
  6. Logical perspectives on the foundations of probability
  7. Subharmonic solutions for a class of predator-prey models with degenerate weights in periodic environments
  8. A non-smooth Brezis-Oswald uniqueness result
  9. Luenberger compensator theory for heat-Kelvin-Voigt-damped-structure interaction models with interface/boundary feedback controls
  10. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part II)
  11. Positive solution for a nonlocal problem with strong singular nonlinearity
  12. Analysis of solutions for the fractional differential equation with Hadamard-type
  13. Hilfer proportional nonlocal fractional integro-multipoint boundary value problems
  14. A comprehensive review on fractional-order optimal control problem and its solution
  15. The θ-derivative as unifying framework of a class of derivatives
  16. Review Articles
  17. On the use of L-functionals in regression models
  18. Minimal-time problems for linear control systems on homogeneous spaces of low-dimensional solvable nonnilpotent Lie groups
  19. Regular Articles
  20. Existence and multiplicity of solutions for a new p(x)-Kirchhoff problem with variable exponents
  21. An extension of the Hermite-Hadamard inequality for a power of a convex function
  22. Existence and multiplicity of solutions for a fourth-order differential system with instantaneous and non-instantaneous impulses
  23. Relay fusion frames in Banach spaces
  24. Refined ratio monotonicity of the coordinator polynomials of the root lattice of type Bn
  25. On the uniqueness of limit cycles for generalized Liénard systems
  26. A derivative-Hilbert operator acting on Dirichlet spaces
  27. Scheduling equal-length jobs with arbitrary sizes on uniform parallel batch machines
  28. Solutions to a modified gauged Schrödinger equation with Choquard type nonlinearity
  29. A symbolic approach to multiple Hurwitz zeta values at non-positive integers
  30. Some results on the value distribution of differential polynomials
  31. Lucas non-Wieferich primes in arithmetic progressions and the abc conjecture
  32. Scattering properties of Sturm-Liouville equations with sign-alternating weight and transmission condition at turning point
  33. Some results for a p(x)-Kirchhoff type variation-inequality problems in non-divergence form
  34. Homotopy cartesian squares in extriangulated categories
  35. A unified perspective on some autocorrelation measures in different fields: A note
  36. Total Roman domination on the digraphs
  37. Well-posedness for bilevel vector equilibrium problems with variable domination structures
  38. Binet's second formula, Hermite's generalization, and two related identities
  39. Non-solid cone b-metric spaces over Banach algebras and fixed point results of contractions with vector-valued coefficients
  40. Multidimensional sampling-Kantorovich operators in BV-spaces
  41. A self-adaptive inertial extragradient method for a class of split pseudomonotone variational inequality problems
  42. Convergence properties for coordinatewise asymptotically negatively associated random vectors in Hilbert space
  43. Relating the super domination and 2-domination numbers in cactus graphs
  44. Compatibility of the method of brackets with classical integration rules
  45. On the inverse Collatz-Sinogowitz irregularity problem
  46. Positive solutions for boundary value problems of a class of second-order differential equation system
  47. Global analysis and control for a vector-borne epidemic model with multi-edge infection on complex networks
  48. Nonexistence of global solutions to Klein-Gordon equations with variable coefficients power-type nonlinearities
  49. On 2r-ideals in commutative rings with zero-divisors
  50. A comparison of some confidence intervals for a binomial proportion based on a shrinkage estimator
  51. The construction of nuclei for normal constituents of Bπ-characters
  52. Weak solution of non-Newtonian polytropic variational inequality in fresh agricultural product supply chain problem
  53. Mean square exponential stability of stochastic function differential equations in the G-framework
  54. Commutators of Hardy-Littlewood operators on p-adic function spaces with variable exponents
  55. Solitons for the coupled matrix nonlinear Schrödinger-type equations and the related Schrödinger flow
  56. The dual index and dual core generalized inverse
  57. Study on Birkhoff orthogonality and symmetry of matrix operators
  58. Uniqueness theorems of the Hahn difference operator of entire function with a Picard exceptional value
  59. Estimates for certain class of rough generalized Marcinkiewicz functions along submanifolds
  60. On semigroups of transformations that preserve a double direction equivalence
  61. Positive solutions for discrete Minkowski curvature systems of the Lane-Emden type
  62. A multigrid discretization scheme based on the shifted inverse iteration for the Steklov eigenvalue problem in inverse scattering
  63. Existence and nonexistence of solutions for elliptic problems with multiple critical exponents
  64. Interpolation inequalities in generalized Orlicz-Sobolev spaces and applications
  65. General Randić indices of a graph and its line graph
  66. On functional reproducing kernels
  67. On the Waring-Goldbach problem for two squares and four cubes
  68. Singular moduli of rth Roots of modular functions
  69. Classification of self-adjoint domains of odd-order differential operators with matrix theory
  70. On the convergence, stability and data dependence results of the JK iteration process in Banach spaces
  71. Hardy spaces associated with some anisotropic mixed-norm Herz spaces and their applications
  72. Remarks on hyponormal Toeplitz operators with nonharmonic symbols
  73. Complete decomposition of the generalized quaternion groups
  74. Injective and coherent endomorphism rings relative to some matrices
  75. Finite spectrum of fourth-order boundary value problems with boundary and transmission conditions dependent on the spectral parameter
  76. Continued fractions related to a group of linear fractional transformations
  77. Multiplicity of solutions for a class of critical Schrödinger-Poisson systems on the Heisenberg group
  78. Approximate controllability for a stochastic elastic system with structural damping and infinite delay
  79. On extremal cacti with respect to the first degree-based entropy
  80. Compression with wildcards: All exact or all minimal hitting sets
  81. Existence and multiplicity of solutions for a class of p-Kirchhoff-type equation RN
  82. Geometric classifications of k-almost Ricci solitons admitting paracontact metrices
  83. Positive periodic solutions for discrete time-delay hematopoiesis model with impulses
  84. On Hermite-Hadamard-type inequalities for systems of partial differential inequalities in the plane
  85. Existence of solutions for semilinear retarded equations with non-instantaneous impulses, non-local conditions, and infinite delay
  86. On the quadratic residues and their distribution properties
  87. On average theta functions of certain quadratic forms as sums of Eisenstein series
  88. Connected component of positive solutions for one-dimensional p-Laplacian problem with a singular weight
  89. Some identities of degenerate harmonic and degenerate hyperharmonic numbers arising from umbral calculus
  90. Mean ergodic theorems for a sequence of nonexpansive mappings in complete CAT(0) spaces and its applications
  91. On some spaces via topological ideals
  92. Linear maps preserving equivalence or asymptotic equivalence on Banach space
  93. Well-posedness and stability analysis for Timoshenko beam system with Coleman-Gurtin's and Gurtin-Pipkin's thermal laws
  94. On a class of stochastic differential equations driven by the generalized stochastic mixed variational inequalities
  95. Entire solutions of two certain Fermat-type ordinary differential equations
  96. Generalized Lie n-derivations on arbitrary triangular algebras
  97. Markov decision processes approximation with coupled dynamics via Markov deterministic control systems
  98. Notes on pseudodifferential operators commutators and Lipschitz functions
  99. On Graham partitions twisted by the Legendre symbol
  100. Strong limit of processes constructed from a renewal process
  101. Construction of analytical solutions to systems of two stochastic differential equations
  102. Two-distance vertex-distinguishing index of sparse graphs
  103. Regularity and abundance on semigroups of partial transformations with invariant set
  104. Liouville theorems for Kirchhoff-type parabolic equations and system on the Heisenberg group
  105. Spin(8,C)-Higgs pairs over a compact Riemann surface
  106. Properties of locally semi-compact Ir-topological groups
  107. Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2
  108. Ordering stability of Nash equilibria for a class of differential games
  109. A new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order
  110. About a dubious proof of a correct result about closed Newton Cotes error formulas
  111. Ricci ϕ-invariance on almost cosymplectic three-manifolds
  112. Schur-power convexity of integral mean for convex functions on the coordinates
  113. A characterization of a ∼ admissible congruence on a weakly type B semigroup
  114. On Bohr's inequality for special subclasses of stable starlike harmonic mappings
  115. Properties of meromorphic solutions of first-order differential-difference equations
  116. A double-phase eigenvalue problem with large exponents
  117. On the number of perfect matchings in random polygonal chains
  118. Evolutoids and pedaloids of frontals on timelike surfaces
  119. A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine
  120. The 𝔪-WG° inverse in the Minkowski space
  121. Stability result for Lord Shulman swelling porous thermo-elastic soils with distributed delay term
  122. Approximate solvability method for nonlocal impulsive evolution equation
  123. Construction of a functional by a given second-order Ito stochastic equation
  124. Global well-posedness of initial-boundary value problem of fifth-order KdV equation posed on finite interval
  125. On pomonoid of partial transformations of a poset
  126. New fractional integral inequalities via Euler's beta function
  127. An efficient Legendre-Galerkin approximation for the fourth-order equation with singular potential and SSP boundary condition
  128. Eigenfunctions in Finsler Gaussian solitons
  129. On a blow-up criterion for solution of 3D fractional Navier-Stokes-Coriolis equations in Lei-Lin-Gevrey spaces
  130. Some estimates for commutators of sharp maximal function on the p-adic Lebesgue spaces
  131. A preconditioned iterative method for coupled fractional partial differential equation in European option pricing
  132. A digital Jordan surface theorem with respect to a graph connectedness
  133. A quasi-boundary value regularization method for the spherically symmetric backward heat conduction problem
  134. The structure fault tolerance of burnt pancake networks
  135. Average value of the divisor class numbers of real cubic function fields
  136. Uniqueness of exponential polynomials
  137. An application of Hayashi's inequality in numerical integration
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 10.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/math-2023-0612/html
Scroll to top button