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Generalized Munn rings

  • Junying Guo and Xiaojiang Guo EMAIL logo
Published/Copyright: October 6, 2022
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Open Mathematics
From the journal Open Mathematics Volume 20 Issue 1

Abstract

Generalized Munn rings exist extensively in the theory of rings. The aim of this note is to answer when a generalized Munn ring is primitive (semiprimitive, semiprime and prime, respectively). Sufficient and necessary conditions are obtained for a generalized Munn ring with a regular sandwich matrix to be primitive (semiprimitive, semiprime and prime, respectively). Also, we obtain sufficient and necessary conditions for a Munn ring over principal ideal domains to be prime (semiprime, respectively). Our results can be regarded as the generalizations of the famous result in the theory of rings that for a ring R , R is primitive (semiprimitive and semiprime, respectively) if and only if so is M n ( R ) . As applications of our results, we consider the primeness and the primitivity of generalized matrix rings and generalized path algebras. In particular, it is proved that a path algebra is a semiprime if and only if it is semiprimitive.

Keywords: generalized Munn ring; generalized path algebra; semiprimitive ring; semiprime ring; generalized matrix ring
MSC 2010: 16S36; 15A30; 05C25

1 Introduction

Throughout this note, we shall use the standard notions and notations, and each of the considered rings is associative but has possibly no identity.

The class of generalized matrix rings has been extensively studied. Examples of generalized matrix rings include piecewise domains (see [1]), incidence algebras of directed graphs (see, [2,3]), structural matrix rings (see [4] and subsequent papers), endomorphism rings, and Morita context rings. Sands [5] observed that if [ S , V , W , T ] is a Morita context, then

S V W T

is a ring. These Morita context rings are precisely generalized matrix rings with idempotent sets E such that ∣ E ∣ = 2 , and they have been widely studied. In particular, we note Amitsur’s paper [6], the survey paper [7], McConnell and Robson’s treatment [8] and Müller’s computation of the maximal quotient ring [9]. Indeed, cellular algebras, affine cellular algebras and standardly based algebras there exist some “local” structures of generalized Munn algebras (for example, see [10,11,12]).

Brown [13] considered generalized matrix algebras of finite dimension over a field of characteristic 0. He proved that such a generalized matrix algebra is either simple or nonsemisimple and simple modulo its radical, and it is simple if and only if it possesses an identity. Sands [5] gave the prime radical of generalized matrix rings with a finite idempotent set. Zhang [14] considered the prime radical of the general case. In 1989, Wauters and Jespers [15] determined when a generalized matrix ring with a finite idempotent set is semiprime. Classical quotient rings of generalized matrix rings with finite idempotent sets had been attracting due attention. There are a series of papers on this field (see [16,17,18]).

Indeed, any generalized matrix ring can be viewed as a subring of some generalized Munn ring. It is natural to research generalized Munn rings. Li [19] considered the regularity of Munn rings. The main aim of this study is to answer when a generalized Munn ring is semiprimitive (semiprime and prime, respectively). A sufficient and necessary condition is established for a generalized Munn ring with a regular sandwich matrix to be primitive (semiprimitive, semiprime and prime, respectively) (Theorems 2.5 and 2.7). Moreover, we answer when a Munn ring over a principal ideal domain is semiprime (prime, respectively) (Proposition 2.11). In Section 3, we determine when a generalized matrix ring is primitive (semiprimitive, semiprime and prime, respectively) (Theorems 3.2 and 3.3). Finally, we consider the primeness and the primitivity of generalized path algebras. It is proved that for a quiver Q and a field K , the path algebra K ( Q ) is semiprime if and only if the path-connected quiver of Q ¯ P C is the disjoint union of complete quivers; if and only if K ( Q ) is semiprimitive (Theorem 4.9). And, K ( Q ) is prime if and only if Q ¯ P C is a complete quiver (Theorem 4.10).

2 Generalized Munn rings

The aim of this section is to consider the primeness and the primitivity of generalized Munn rings.

2.1 Regular sandwich matrices

To begin with, we give the notion of regular matrices.

Definition 2.1

Let A be a ring and X = ( x i m ) be a generalized I × M matrix over A . A nonzero entry x i m of X is called a unit entry of X if there exists a nonzero idempotent e ∈ A such that x i m is a unit in e A e .

Notice that a group has exactly one idempotent, which is just the identity of the group. This means that in Definition 2.1, the idempotent e is indeed unique. So, the unique idempotent e in Definition 2.1 is denoted by x i m ◇ .

Definition 2.2

Let A be a ring. An I × M matrix X = ( x i m ) over A is said regular in A if the following conditions hold:

  1. For any i ∈ I , there exists m ∈ M such that x i m is a unit entry of X .

  2. For any n ∈ M , there exists j ∈ I such that x j n is a unit entry of X .

  3. If x i 0 m 0 is a unit entry of X , then

    1. x i 0 m 0 ◇ x i 0 m = x i 0 m for any m ∈ M ;

    2. x i m 0 x i 0 m 0 ◇ = x i m 0 for any i ∈ I .

By definition, any m × m matrix without zero rows and zero columns are regular in the field C of complex numbers. Also, for a ring A with unity, any I × M matrix over A , in which each row and each column contains at least one unit of A , must be regular in A .

Let M , I be nonempty sets, and A an associative ring and Q = ( q m i ) a generalized M × I matrix over A . Consider the set M ( A , I , M ) consisting of all generalized I × M matrices over A with only finite nonzero entries, such an I × M matrix is usually said to be bounded. For C = ( a i m ) , D = ( b i m ) ∈ M ( A , I , M ) , define

C + D = ( e i m ) , where e i m = c i m + d i m for i ∈ I , m ∈ M ; C ∘ D = C Q D , where the product on the right side is the product of matrices; λ C = ( λ c i m ) for λ ∈ R .

By definition, a routine calculation shows that with these operations, M ( A , I , M ) is an associative ring.

Definition 2.3

The above ring M ( A , I , M ) is called a generalized Munn ring A with the sandwich matrix Q , in notation, M ( A , I , M ; Q ) .

If I is finite, then we identity it with the set { 1 , 2 , … , i } , where i is the cardinality of I , and we write M ( A , I , M ; Q ) as M ( A , i , M ; Q ) . Similarly, the notation M ( A , I , m ; Q ) is used if ∣ M ∣ = m < ∞ . Denote by M m , n ( A ) the set of all m × n matrices over A .

Recall from [20] that the generalized Munn ring M ( A , m , n ; Q ) is called the Munn m × n matrix ring over A with sandwich matrix Q . It is obvious that M n ( A ) is the Munn n × n matrix ring over A with sandwich matrix Δ , where Δ is the unit matrix; that is, the diagonal matrix each of whose entries in the diagonal positions is the unity of A .

Definition 2.4

Let T be a nonempty subset of M ( A , I , M ) . An M × I matrix X = ( x m i ) over A is said to be cancellable in T if for any nonzero element Y ∈ T , Y X and X Y are neither zero, where Y X and X Y are usual matrix products.

Evidently, for a ring A with identity 1, the I × I unit matrix Δ is cancellable in any subset of M ( A , I , I ) . And, any invertible n × n matrix must be cancellable in M n ( A ) , but not all of cancellable matrices in M n ( A ) are invertible in the matrix algebra.

Example. Let Z be the ring of integers. It is easy to check that the matrix

A = 2 0 0 4

is cancellable in the matrix ring M 2 ( Z ) . But A is not invertible in M 2 ( Z ) .

For convenience, we denote

( a ) i m : the generalized I × M matrix with a in the ( i , m ) position and 0 elsewhere; ( ℬ ) i m : the set { ( b ) i m : b ∈ ℬ } for ℬ ⊆ A ; ( a i m ) i ∈ J , m ∈ N : the generalized I × M matrix with a i m in the ( i , m ) position for i ∈ J , m ∈ N and 0 elsewhere. Especially, if N = { n } , we simply write m ∈ N as m = n , and the similar sign for the case : ∣ I ∣ = 1 . rad ( A ) : the Jacobson radical of the ring A .

We now arrive at our main results of this note.

Theorem 2.5

Let M = M ( A , I , M ; Q ) be a generalized Munn ring. If Q is regular in A , then M ( A , I , M ; Q ) is semiprime (semiprimitive, respectively) if and only if the following conditions are satisfied:

  1. Q is cancellable in M ( A , I , M ) ;

  2. for any unit entries q m i , q n j of Q , if x is a nonzero element of q m i ◇ A q n j ◇ , then q n j ◇ A q m i ◇ x ≠ 0 and x q n j ◇ A q m i ◇ ≠ 0 ;

  3. for any unit entry q m i of Q , q m i ◇ A q m i ◇ is semiprime (semiprimitive, respectively).

Proof

Let q m i be a unit entry of Q and denote by q m i − 1 the inverse of q m i in q m i ◇ A q m i ◇ , obviously q m i ◇ = q m i q m i − 1 = q m i − 1 q m i . Then ( q m i − 1 ) i m is an idempotent of M , and

(2.1) ( q m i − 1 ) i m ∘ M ∘ ( q m i − 1 ) i m = ( q m i ◇ A q m i ◇ ) i m

since q m i is a unit in q m i ◇ A q m i ◇ . A routine calculation shows that the mapping

(2.2) ϕ : ( x ) i m ↦ x q m i is an isomorphism from ( q m i ◇ A q m i ◇ ) i m onto q m i ◇ A q m i ◇ .

(2.5.1) The proof for the semiprime case. If M is semiprime, then by (2.1), ( q m i ◇ A q m i ◇ ) i m is semiprime, so that by ϕ is an isomorphism, q m i ◇ A q m i ◇ is semiprime. It results (iii). To see (i), assume on the contrary that Q is not cancellable in M ( A , I , M ) , then there is a nonzero element X ∈ M such that X Q = 0 or Q X = 0 . Without loss of generality, let X Q = 0 , so that X ∘ U ∘ X = X Q U Q X = 0 for any U ∈ M , whence X ∘ M ∘ X = 0 , contrary to that M is semiprime. Thus, Q is cancellable in M ( A , I , M ) .

We next verify (ii). To the end, we assume contrariwise that there exists a nonzero element x ∈ q m i ◇ A q n j ◇ such that x q n j ◇ A q m i ◇ = 0 . Obviously, x q n j ◇ = x = q m i ◇ x , so that

( x ) i m ∘ M ∘ ( x ) i m ⊆ ( x A x ) i m = ( ( x q n j ◇ ) A ( q m i ◇ x ) ) i m = ( x ( q n j ◇ A q m i ◇ ) x ) i m = 0 ,

contrary to the hypothesis that M is semiprime. Therefore, x q n j ◇ A q m i ◇ ≠ 0 . Similarly, q n j ◇ A q m i ◇ x ≠ 0 . We have now proved that (ii) is valid.

For the converse, we contrariwise let w = ( w i m ) be a nonzero generalized I × M matrix in M such that w ∘ M ∘ w = 0 . Because Q is cancellable, Q w Q = ( u m i ) m ∈ M , i ∈ I ≠ 0 , and we assume that u m 0 i 0 ≠ 0 . If w Q = ( v i j ) i , j ∈ I , then

(2.3) u m 0 i 0 = ∑ j ∈ I q m 0 j v j i 0 .

When Q is regular, there is j 0 ∈ I such that q m 0 j 0 is a unit entry of Q . It follows that q m 0 j 0 ◇ q m 0 j = q m 0 j . Now by (2.3),

u m 0 i 0 = ∑ j ∈ I q m 0 j 0 ◇ q m 0 j v j i 0 = q m 0 j 0 ◇ ∑ j ∈ I q m 0 j v j i 0 = q m 0 j 0 ◇ u m 0 i 0 ,

and similarly, there exists a unit entry q n 0 i 0 of Q such that u m 0 i 0 q n 0 i 0 ◇ = u m 0 i 0 . Therefore, u m 0 i 0 ∈ q m 0 j 0 ◇ A q n 0 i 0 ◇ . Furthermore, by (ii), there is x ∈ q n 0 i 0 ◇ A q m 0 j 0 ◇ such that 0 ≠ u m 0 i 0 x ∈ q m 0 j 0 ◇ A q m 0 j 0 ◇ . Clearly,

(2.4) q m 0 j 0 ◇ u m 0 i 0 x = u m 0 i 0 x = ( u m 0 i 0 x ) q m 0 j 0 ◇ .

It follows that

(2.5) u ≔ ( q m 0 j 0 ◇ ) j 0 m 0 Q w Q ( x ) i 0 m 0 = ( q m 0 j 0 ◇ ) j 0 m 0 ( u m 0 i 0 ) m 0 i 0 ( x ) i 0 m 0 = ( q m 0 j 0 ◇ u m 0 i 0 x ) j 0 m 0 = ( u m 0 i 0 x ) j 0 m 0 = ( q m 0 j 0 ◇ ) j 0 m 0 ∘ w ∘ ( x ) i 0 m 0 ≠ 0 .

Now

( u m 0 i 0 x ⋅ q m 0 j 0 ◇ A q m 0 j 0 ◇ ⋅ u m 0 i 0 x ) j 0 m 0 = ( u m 0 i 0 x q m 0 j 0 ◇ ⋅ A ⋅ q m 0 j 0 ◇ u m 0 i 0 x ) j 0 m 0 = ( u m 0 i 0 x ) j 0 m 0 Q ( q m 0 j 0 − 1 A q m 0 j 0 − 1 ) j 0 m 0 Q ( u m 0 i 0 x ) j 0 m 0 ⊆ ( u m 0 i 0 x ) j 0 m 0 Q M Q ( u m 0 i 0 x ) j 0 m 0 = ( u m 0 i 0 x ) j 0 m 0 ∘ M ∘ ( u m 0 i 0 x ) j 0 m 0 = ( ( q m 0 j 0 ◇ ) j 0 m 0 ∘ w ∘ ( x ) j 0 m 0 ) ∘ M ∘ ( ( q m 0 j 0 ◇ ) j 0 m 0 ∘ w ∘ ( x ) j 0 m 0 ) ⊆ ( q m 0 j 0 ◇ ) j 0 m 0 ∘ ( w ∘ M ∘ w ) ∘ ( x ) j 0 m 0 = 0 ,

so that u m 0 i 0 x ⋅ q m 0 j 0 ◇ A q m 0 j 0 ◇ ⋅ u m 0 i 0 x = 0 . This is contrary to that q m 0 j 0 ◇ A q m 0 j 0 ◇ is semiprime. Consequently, M is semiprime.

(2.5.2) The proof for the semiprimive case. It is well known that for a semiprimitive algebra A and an idempotent e ∈ A , e A e is still semiprimitive. So, the same reason as in (2.5.1) shows that the “if” part is valid. With notations in (2.5.1), if w ∈ rad ( M ) \ { 0 } , then by (2.5), 0 ≠ ( u m 0 i 0 x ) j 0 m 0 ∈ rad ( M ) . Notice that ( q m 0 j 0 − 1 ) j 0 m 0 is an idempotent of M , we can obtain that

( u m 0 i 0 x ) j 0 m 0 = ( q m 0 j 0 ◇ u m 0 i 0 x q m 0 j 0 ◇ ) j 0 m 0 ( by (2.3) ) = ( q m 0 j 0 − 1 ) j 0 m 0 ∘ ( u m 0 i 0 x ) j 0 m 0 ∘ ( q m 0 j 0 − 1 ) j 0 m 0 ∈ ( q m 0 j 0 − 1 ) j 0 m 0 ∘ rad ( M ) ∘ ( q m 0 j 0 − 1 ) j 0 m 0 = rad ( ( q m 0 j 0 − 1 ) j 0 m 0 ∘ M ∘ ( q m 0 j 0 − 1 ) j 0 m 0 ) = rad ( ( q m 0 j 0 − 1 A q m 0 j 0 − 1 ) j 0 m 0 ) = rad ( ( q m 0 j 0 ◇ A q m 0 j 0 ◇ ) j 0 m 0 ) .

This means that ( q m 0 j 0 ◇ A q m 0 j 0 ◇ ) j 0 m 0 is not semiprimitive. Now by (2.2), we observe that q m 0 j 0 ◇ A q m 0 j 0 ◇ is not semiprimitive. This is contrary to the hypothesis. It results the “only if” part.□

Remark 2.6

Let us turn back to the proof of Theorem 2.5. In (2.5.1), the proof of Condition (i) in the direct part has indeed proved that if M ( A , I , M ; Q ) is semiprime, then Q is cancellable in M ( A , I , M ) .

Theorem 2.7

Let M = M ( A , I , M ; Q ) be a generalized Munn ring. If Q is regular in A , then M ( A , I , M ; Q ) is prime (primitive, respectively) if and only if the following conditions are satisfied:

  1. Q is cancellable in M ( A , I , M ) ;

  2. for any unit entries q m i , q n j , q r k , q s l of Q , if x and y are nonzero elements of q m i ◇ A q n j ◇ and q r k ◇ A q s l ◇ , respectively, then x q n j ◇ A q r k ◇ y ≠ 0 ;

  3. for any unit entry q m i of Q , q m i ◇ A q m i ◇ is prime (primitive, respectively).

Proof

(2.7.1) The proof for the prime case. If M is prime, then by (2.1), ( q m i ◇ A q m i ◇ ) i m is prime, so that by ϕ is an isomorphism, q m i ◇ A q m i ◇ is prime. By Remark 2.6, Q is cancellable in M ( A , I , M ) .

We next verify (ii). We contrariwise let x ∈ q m i ◇ A q n j ◇ \ { 0 } , y ∈ q r k ◇ A q s l ◇ \ { 0 } such that x q n j ◇ A q r k ◇ y = 0 . Then, x q n j ◇ = x and q r k ◇ y = y . Moreover,

( x ) i m ∘ M ∘ ( y ) i m = ( x A y ) i m = ( x ⋅ q n j ◇ A q r k ◇ ⋅ y ) i m = ( 0 ) i m = 0 .

It is contrary to the hypothesis that M is prime. We have now proved the necessity.

To see the converse part, we assume conversely that there exist nonzero elements A , B ∈ M such that A ∘ M ∘ B = 0 . It is not difficult to see that Condition (ii) in Theorem 2.7 implies Condition (ii) in Theorem 2.5. Indeed, by (ii), for any x ∈ q m i ◇ A q n j ◇ , x q n j ◇ A q m i ◇ x ≠ 0 , so that x q n j ◇ A q n i ◇ ≠ 0 ; similarly, q n j ◇ A q m i ◇ x ≠ 0 , and it results immediately in Condition (ii) in Theorem 2.5. Now by (2.5.1) (precisely, see (2.4) and (2.5)), there are m , n ∈ M , j , k ∈ I and C 1 , C 2 , D 1 , D 2 ∈ M such that

  1. q m j , q n k are unit entries of Q ; and

  2. C 1 ∘ A ∘ D 1 = ( a ) j m and C 2 ∘ B ∘ D 2 = ( b ) k n , where a and b are nonzero elements in q m j ◇ A q m j ◇ and q n k ◇ A q n k ◇ , respectively.

Furthermore by (ii), we have u ∈ q m j ◇ A q n k ◇ such that a u b ≠ 0 . Obviously, q m j ◇ u = u = u q n k ◇ .

Compute

(2.6) ( a u b ) j n ∘ M ∘ ( a u b ) j n = ( a q m j ◇ u b ) j n ∘ M ∘ ( a u q n k ◇ b ) j n = ( a q m j q m j − 1 u b ) j n ∘ M ∘ ( a u q n k − 1 q n k b ) j n = ( a ) j m ∘ ( q m j − 1 u b ) j n ∘ M ∘ ( a u q n k − 1 ) j n ∘ ( b ) k n ⊆ ( a ) j m ∘ M ∘ ( b ) k n = C 1 ∘ A ∘ D 1 ∘ M ∘ C 2 ∘ B ∘ D 2 ⊆ C 1 ∘ A ∘ M ∘ B ∘ D 2 = 0 .

This shows that M is not a semiprime ring. But by Theorem 2.5, M is semiprime. It is a contradiction. Therefore, M is prime.

(2.7.2) The proof for the primitive case. By the well-known result (for example, see [21, Ex. 10, p. 339]): for any primitive algebra A and any idempotent e in A , e A e is still primitive, and since any primitive algebra is prime, a similar argument as in (2.5.1) can verify the “if” part. For the converse, if given conditions hold, then by (2.7.1), M is prime. The rest follows from a famous result of Lanahi et al. [22] showed that for a prime ring R , if e is a nonzero idempotent in R , then R is primitive if and only if e R e is primitive.□

Based on Theorems 2.5 and 2.7, we may prove the following proposition.

Proposition 2.8

Let M ( A , I , M ; Q ) be a generalized Munn ring. Assume that

  1. A has a unity;

  2. each row and each column of Q contains at least one unit of A .

Then M ( A , I , M ; Q ) is prime (semiprime, primtive and semiprimitive, respectively) if and only if the following conditions are satisfied:

  1. Q is cancellable in M ( A , I , M ) ;

  2. A is prime (semiprime, primtive and semiprimitive, respectively).

Proof

By definition, q m i ◇ is the unity of A for any unit entry q m i of Q satisfying Condition (2); in this case, q m i ◇ A q m i ◇ = A . Obviously, Q is regular in A .

Let q m i be an arbitrary unit entry of Q . By Condition (2), there is an entry q m i 0 of Q such that q m i 0 is a unit in A . But Q is regular in A , so q m i ◇ q m i 0 = q m i 0 , and it follows that q m i ◇ must be the unity of A . We have now proved that any unit entry of Q is a unit in A . This shows that Condition (ii) in Theorem 2.5 are satisfied and that Condition (ii) in Theorem 2.7 is satisfied whenever A is prime.

The rest follows immediately from Theorems 2.5 and 2.7.□

For a ring A with unity, denote by Δ the generalized I × I matrix over A each of whose entries in the diagonal positions is the unity of A and 0 elsewhere. Obviously, Δ is cancellable in M ( A , I , I ) . It is easy to see that the following corollary is an easy consequence of Proposition 2.8.

Corollary 2.9

Let A be a ring with unity. Then A is prime (semiprime, primitive and semiprimitive, respectively) if and only if for any [for some] nonempty set I , M ( A , I , I ; Δ ) is prime (semiprime, primtive and semiprimitive, respectively).

Let us turn back to the proof of Theorem 2.7. Assume now that M is semiprime and the condition:

  1. If q m j , q n k are unit entries of Q , then a q m j ◇ A q n k ◇ b ≠ 0 for any nonzero elements a ∈ q m j ◇ A q m j ◇ , b ∈ q n k ◇ A q n k ◇ .

In this case, u in (2.5) exists in M . Moreover, we can derive Conditions (a) and (b) in the proof of Theorem 2.7, and whence (2.6). So, we have indeed proved the following theorem.

Theorem 2.10

Let M ( A , I , M ; Q ) be a generalized Munn ring. If Q is regular in A , then M is prime if and only if M is semiprime and (PM) is satisfied.

Comparing with Theorem 2.10, it raises a natural conjecture as follows:

Conjecture 2.11

Let M ( A , I , M ; Q ) be a generalized Munn ring, and assume that Q is regular in A . Then the following conditions are equivalent:

  1. M is primitive;

  2. M is semiprimitive and (PM) is satisfied;

  3. M is both semiprimitive and prime.

2.2 Principal ideal domains

In this subsection, we study the primeness of Munn rings over a principal ideal domain. We first provide one property of cancellable matrices over a principal ideal domain.

Lemma 2.12

Let A be a principal ideal domain, and Q a m × n matrix over A . Then Q is cancellable in M ( A , n , m ) if and only if m = n = r Q , where r Q is the rank of Q .

Proof

(Necessity). Assume that Q is cancellable in M ( A , I , M ) . We suppose contrariwise that m = n = r Q is not valid. By [23, Proposition III.2.11], there exist invertible matrices U , V such that

U Q V = D 0 0 0 ,

where D is a diagonal r Q × r Q matrix with nonzero diagonal entries. So, there exists a nonzero matrix A 22 over A such that

0 0 0 A 22 U Q V = 0 .

Moreover,

0 0 0 A 22 U Q = 0 ,

so that

0 0 0 A 22 = 0

since, by hypothesis, Q is cancellable in M ( A , I , M ) . It follows that A 22 = 0 . It is a contradiction. Thus, m = r Q = n .

(Sufficiency). If m = r Q = n , then by [23, Proposition III.2.11], there exist invertible matrices U , V such that U Q V = diag ( d 1 , d 2 , … , d m ) , where d i ≠ 0 for i = 1 , 2 , … , m . For any X = ( x i j ) ∈ M n ( A ) , we have

X Q = 0 ⇔ ( X U − 1 ) U Q V = ( y i j d j ) = 0 , where X U − 1 = ( y i j ) ; ⇔ y i j d j = 0 for i , j = 1 , 2 , … , n ; ⇔ y i j = 0 for i , j = 1 , 2 , … , n ; ⇔ X U − 1 = 0 ; ⇔ X = 0 ,

and similarly, Q X = 0 if and only if X = 0 . Therefore, Q is cancellable in M ( A , m , n ) .□

Proposition 2.13

Let M ( A , m , n ; Q ) be a Munn ring. If A is a principal ideal domain with unity 1, then the following conditions are equivalent:

  1. M is semiprime;

  2. m = r Q = n ;

  3. M is prime.

Proof

(i) ⇒ (ii). By Remark 2.6, Q is cancellable in M ( A , m , n ) . Now Lemma 2.12 results (ii).

(ii) ⇒ (iii). Let A 1 and A 2 be an arbitrary nonzero n × n matrices over A . By [23, Proposition III.2.11], there exist invertible matrices U i , V i , i = 1 , 2 such that

U i A i V i = diag ( d 1 ( i ) , d 2 ( i ) , … , d m i ( i ) , 0 , … , 0 ) ,

where d k ( i ) ≠ 0 for any 1 ≤ k ≤ m i . Let U and V be invertible matrices such that Q = U diag ( c 1 , c 2 , … , c n ) V , where c j ≠ 0 for j = 1 , 2 , … , n . Because

V 1 − 1 U = ( x i j ) , V U 2 − 1 = ( y i j )

are both invertible, there exist 1 ≤ j 0 , i 0 ≤ n such that x 1 j 0 ≠ 0 , y i 0 1 ≠ 0 . Compute

( 1 ) 11 U 1 A 1 Q V − 1 ( 1 ) j 0 i 0 U − 1 Q A 2 V 2 ( 1 ) 11 = ( 1 ) 11 ( U 1 A 1 V 1 ) ( V 1 − 1 U ) diag ( c 1 , … , c n ) ( 1 ) j 0 i 0 ( U − 1 Q V − 1 ) ( V U 2 − 1 ) ( U 2 A 2 V 2 ) ( 1 ) 11 = ( 1 ) 11 diag ( d 1 ( 1 ) , … , d m 1 ( 1 ) , 0 ⋯ , 0 ) ( x i j ) diag ( c 1 , c 2 , … , c n ) ⋅ ( 1 ) j 0 i 0 diag ( c 1 , c 2 , … , c n ) ( y i j ) diag ( d 1 ( 2 ) , … , d m 2 ( 2 ) , 0 ⋯ , 0 ) ( 1 ) 11 = ( d 1 ( 1 ) x 1 j 0 c j 0 c i 0 y i 0 1 d 1 ( 2 ) ) 11 ≠ 0 ,

so that the entry in the ( 1 , 1 ) position of

U 1 A 1 Q V − 1 ( 1 ) j 0 i 0 U − 1 Q A 2 V 2 = U 1 A 1 ∘ V − 1 ( 1 ) j 0 i 0 U − 1 ∘ A 2 V 2

is equal to the nonzero element d 1 ( 1 ) x 1 j 0 c j 0 c i 0 y i 0 1 d 1 ( 2 ) . It follows that A 1 ∘ V − 1 ( 1 ) j 0 i 0 U − 1 ∘ A 2 ≠ 0 since A is a principal ideal domain, giving d 1 ( 1 ) x 1 j 0 c j 0 c i 0 y i 0 1 d 1 ( 2 ) ≠ 0 . Thus, A 1 ∘ M ∘ A 2 ≠ 0 and whence M is prime.

(iii) ⇒ (i). It is obvious.□

Notice that for a principal ideal domain, the unity is the only nonzero idempotent. We observe that a generalized matrix over a principal ideal domain is regular if and only if each of its rows and each of its columns contain at least one unit. By Propositions 2.8 and 2.13, the following corollary is immediate from that any domain is prime.

Corollary 2.14

Let M ( A , m , n ; Q ) be a Munn ring. If

  1. A is a principal ideal domain;

  2. Q is regular in A ,

then M is prime if and only if m = r Q = n .

3 Generalized matrix rings

In this section, we shall consider the primeness and the primitivity of generalized matrix rings. We first recall the definition of generalized matrix rings.

Let I be a nonempty set. For any i , j , l ∈ I , let A i i be a ring with identity 1 i , and A i j a unitary ( A i i , A j j ) -bimodule. Assume that there is a module homomorphism μ i j l : A i j × A j l into A i l , written μ i j l ( x , y ) = x y , for any i , j , l ∈ I . If the following conditions hold:

  1. ( x + y ) z = x z + y z , w ( x + y ) = w x + w y ;

  2. w ( x z ) = ( w x ) z ,

for any x , y ∈ A i j , z ∈ A j l , w ∈ A l i , then the triple ( A i j , I , μ i j l ) is called a Γ -system with index I .

Given a Γ -system ( A i j , I , μ i j l ) , let Gℳ = Gℳ ( A i j , I , μ i j l ) be the external direct sum of { A i j : i , j ∈ I } . We shall use { x i j } to denote the external direct sum of x i j with i , j ∈ I . Now we define the multiplication in Gℳ as

x y = ∑ k x i k y k j ,

where x = { x i j } and y = { y i j } . It is easy to check that Gℳ is a ring (possibly without unity). We call Gℳ a generalized matrix ring, or a gm ring for short, written Gℳ ( A i j , I , μ i j k ) or Gℳ for short. If e i i is a nonzero element of A i i satisfying that e i i x = x = x e j j for all x ∈ A i j , then the set { e i i : i ∈ I } is called a generalized matrix unit of the Γ -system ( A i j , I , μ i j u ) (for example, see [14]).

In what follows, we still write the element x = { x i j } satisfying that x i j = 0 if i ≠ i 0 , j ≠ j 0 and x i 0 j 0 = u as { u } i 0 j 0 , especially, write { 1 i 0 } i 0 i 0 = 1 i 0 . Also, we use { A } i j to stand for the set { { a } i j : a ∈ A } . And, we write x = { x i j } i ∈ A , j ∈ B if x i j = 0 whenever i ∈ I \ A or j ∈ I \ B . It is easy to check that the set { 1 i : i ∈ I } is a generalized matrix unit of the Γ -system ( A i j , I , μ i j u ) .

Proposition 3.1

The generalized matrix ring Gℳ ( A i j , I , μ i j k ) is a subring of the generalized Munn ring M ( Gℳ , I , I ; Ξ ) , where Ξ is the generalized I × I matrix in which any entry in the ( i , i ) position is 1 i , for any i ∈ I , and 0 elsewhere.

Proof

Consider the mapping

ϕ : Gℳ → M ( Gℳ , I , I ; Ξ ) ; { x i j } ↦ ∑ ( x i j ) i j .

A routine calculation shows that ϕ is an injective homomorphism, and here, we omit the detail.□

We can now describe the main results of this section.

Theorem 3.2

Let Gℳ = Gℳ ( A i j , I , μ i j l ) be a generalized matrix ring. Then Gℳ is semiprime (semiprimitive, respectively) if and only if the following conditions are satisfied:

  1. for any i , j ∈ I , if x is a nonzero element in A i j , then x A j i ≠ 0 and A j i x ≠ 0 ;

  2. for any i ∈ I , A i i is semiprime (semiprimtive, respectively).

Proof

A routine calculation shows that { 1 i } i i is an idempotent, for all i ∈ I , and the mapping φ : { x } i i ↦ x is an isomorphism from { A i i } i i onto A i i . Compute

{ 1 i } i i Gℳ { 1 i } i i = { A i i } i i .

So,

(3.1) { 1 i } i i Gℳ { 1 i } i i ≅ A i i .

(3.2.1) The proof for the semiprime case. If Gℳ is semiprime, then as for all i ∈ I , { 1 i } i i is an idempotent in Gℳ , we obtain that { 1 i } i i Gℳ { 1 i } i i is semiprime, so that by (3.1), A i i is semiprime.

We contrariwise let i 0 , j 0 ∈ I such that A i 0 j 0 ≠ 0 but A j 0 i 0 = 0 . Pick a nonzero element a in A i 0 j 0 . Of course, { a } i 0 j 0 is a nonzero element in Gℳ . Compute

(3.2) { a } i 0 j 0 Gℳ { a } i 0 j 0 = { a } i 0 j 0 { A j 0 i 0 } j 0 i 0 { a } i 0 j 0 = { a A j 0 i 0 a } i 0 j 0 ⊆ { A j 0 i 0 } i 0 j 0 = 0 ,

contrary to the hypothesis that Gℳ is semiprime. Therefore we have now proved that for any i , j ∈ I ,

(3.3) A i j ≠ 0 ⇔ A j i ≠ 0 .

To see (i), we assume contrarily that x is a nonzero element in A i 1 j 1 such that x A j 1 i 1 = 0 . By (3.2), we have { x } i 1 j 1 Gℳ { x } i 1 j 1 = { x A j 1 i 1 x } i 1 j 1 = 0 , contrary to the hypothesis that Gℳ is semiprime. So, x A j i ≠ 0 . Dually, we may prove that A j i x ≠ 0 .

For the converse, assume that given Conditions (i) and (ii) hold. We oppositely let u = { u i j } be a nonzero element in Gℳ such that u Gℳ u = 0 . Notice that

  1. Ξ is regular in Gℳ and cancellable in M ( Gℳ , I , I ) ;

  2. { 1 i } i i , i ∈ I are all unit entries of Ξ . Obviously, { 1 i } i i ◇ = { 1 i } i i and furthermore, { 1 i } i i Gℳ { 1 j } j j = { A i j } i j . Together with Condition (i), it is easy to see that Condition (ii) in Theorem 2.5.

By Theorem 2.5, the generalized Munn ring M ( Gℳ , I , I ; Ξ ) is semiprime. Denote

J = { k ∈ I : u k j ≠ 0 for some j ∈ I , or u i k ≠ 0 for some i ∈ I } .

It is not difficult to check that

  1. ε = ∑ i ∈ J ( 1 i ) i i is an idempotent. Moreover, ε M ( Gℳ , I , I ; Ξ ) ε is semiprime.

  2. ε ϕ ( u ) = ϕ ( u ) = ϕ ( u ) ε , where ϕ has the same meanings as in the proof of Proposition 3.1.

Moreover,

ϕ ( u ) ( ε M ( Gℳ , I , I ; Ξ ) ε ) ϕ ( u ) ⊆ ϕ ( u ) ϕ ( Gℳ ) ϕ ( u ) = ϕ ( u Gℳ u ) = 0 ,

contrary to the foregoing proof that ε M ( Gℳ , I , I ; Ξ ) ε is semiprime. Therefore u = 0 and whence Gℳ is semiprime.

(3.2.2) The proof for the semiprimitive case. Similar as (3.2.1), we may prove the necessity.

For the converse, we contrariwise assume that u is a nonzero element in rad ( Gℳ ) . With notations in (3.2.1), we denote X = { { x i j } i ∈ J , j ∈ J } ⊆ Gℳ . It is easy to see that

  1. X is a subalgebra of Gℳ ;

  2. τ = ∑ i ∈ J { 1 i } i i is an idempotent in Gℳ . Moreover, ϕ ( τ ) = ε , τ u = u = u τ and τ Gℳ τ = X ;

  3. ϕ ( X ) = ε M ( Gℳ , I , I ; Ξ ) ε .

Therefore u ∈ rad ( τ Gℳ τ ) = rad ( X ) . Notice that ϕ is an injective homomorphism. We observe that ε M ( Gℳ , I , I ; Ξ ) ε is isomorphic to X . It follows that

ϕ ( u ) ∈ rad ( ε M ( Gℳ , I , I ; Ξ ) ε ) ,

so that ε M ( Gℳ , I , I ; Ξ ) ε is not semiprimitive. Indeed, by the proof of the converse part in (3.2.1), we can obtain that M ( Gℳ , I , I ; Ξ ) is semiprimitive if for any i ∈ I , A i i is semiprimtive. In this case, ε M ( Gℳ , I , I ; Ξ ) ε is semiprimitive, contrary to the foregoing proof that ε M ( Gℳ , I , I ; Ξ ) ε is not semiprimitive. Consequently, Gℳ is semiprimitive.□

Theorem 3.3

Let Gℳ = Gℳ ( A i j , I , μ i j l ) be a generalized matrix ring. Then Gℳ is prime (primitive, respectively) if and only if the following conditions are satisfied:

  1. for any i , j , k , l ∈ I , if x and y are respectively nonzero elements in A i j and in A k l , then x A j k y ≠ 0 ;

  2. for any i ∈ I , A i i is prime (primtive, respectively).

Proof

(3.3.1) The proof for the prime case. Similar as in (3.2.1), we may prove the necessity.

For the sufficiency, we contrarily assume that u = { u i j } and v = { v k l } are nonzero elements in Gℳ such that u Gℳ v = 0 . Obviously, there are i 0 , j 0 , k 0 , and l 0 ∈ I such that u i 0 j 0 and v k 0 l 0 are neither equal to 0. Further, by Condition (i), there is x ∈ A j 0 k 0 such that u i 0 j 0 x v k 0 l 0 ≠ 0 . By the same reason, we have y ∈ A l 0 i 0 such that u i 0 j 0 x v k 0 l 0 y u i 0 j 0 x v k 0 l 0 ≠ 0 . So that 0 ≠ u i 0 j 0 x v k 0 l 0 y ∈ A i 0 i 0 . Compute

{ u i 0 j 0 x v k 0 l 0 y ⋅ A i 0 i 0 ⋅ u i 0 j 0 x v k 0 l 0 y } i 0 i 0 = { u i 0 j 0 } i 0 j 0 { x v k 0 l 0 y } j 0 i 0 Gℳ { u i 0 j 0 x } i 0 k 0 { v k 0 l 0 } k 0 l 0 { y } l 0 i 0 ⊆ { u i 0 j 0 } i 0 j 0 Gℳ { v k 0 l 0 } k 0 l 0 { y } l 0 i 0 = ( { 1 i 0 } i 0 i 0 u { 1 j 0 } j 0 j 0 ) Gℳ ( { 1 k 0 } k 0 k 0 v { 1 l 0 } l 0 l 0 ) { y } l 0 i 0 ⊆ { 1 i 0 } i 0 i 0 ⋅ u Gℳ v ⋅ { 1 l 0 } l 0 l 0 { y } l 0 i 0 = { 1 i 0 } i 0 i 0 ⋅ 0 ⋅ { 1 l 0 } l 0 l 0 { y } l 0 i 0 = 0 .

It follows that u i 0 j 0 x v k 0 l 0 y ⋅ A i 0 i 0 ⋅ u i 0 j 0 x v k 0 l 0 y = 0 . This means that A i 0 i 0 is not semiprime, contrary to Condition (ii). Therefore, Gℳ is prime.

(3.3.2) The proof for the primitive case. It follows from the proof in (2.5.2).□

4 Generalized path algebras

In this section, we consider the primeness and the primitivity of generalized path algebras. We first provide some results on quivers.

4.1 Quivers

We start with the basic definitions. A quiver Q = ( V , E ) is an oriented graph, where V is the vertex set and E is the arrow set. We denote by S : E → V and T : E → V the mappings, where S ( α ) = i and T ( α ) = j when α : i → j is an arrow from i to j . A path in the quiver Q is an ordered sequence of arrows p = α n ⋯ α 1 with T ( α l ) = S ( α l + 1 ) for 1 < l < n , or the symbol e i for i ∈ V . We call the path e i trivial path and define S ( e i ) = i = T ( e i ) . For a nontrivial path p = α n ⋯ α 1 , we define S ( p ) = S ( α n ) and T ( p ) = T ( α 1 ) . A nontrivial path p = α n ⋯ α 1 is said to be

  1. an oriented cycle if S ( p ) = T ( p ) ;

  2. a loop from i to i if n = 1 and S ( p ) = i = T ( p ) .

Definition 4.1

  1. A quiver G with vertex set V is said to be a complete quiver if for any a , b ∈ V with a ≠ b , there are one arrow from a to b and one arrow from b to a .

  2. Let G 1 and G 2 be quivers with vertex set V 1 and arrow set E 1 , and with vertex set V 2 and arrow set E 2 , respectively. A quiver G is said to be a union of G 1 and G 2 if the vertex set of G is V 1 ⊔ V 2 and the arrow set of G is E 1 ⊔ E 2 . If, in addition, both V 1 ⊔ V 2 and E 1 ⊔ E 2 are disjoint unions, then we shall call G to be a disjoint union of G 1 and G 2 .

By an empty graph, we mean a graph without arrows. Obviously, we have the following observations:

  1. The empty graph is a complete quiver if and only if it it has exactly one vertex.

Also,

  1. Let Q = ( V , E ) be a quiver without loops. Q is a disjoint union of complete quivers if and only if for any a , b ∈ V with a ≠ b , if there is a path from a to b , then there is one arrow from b to a .

Indeed, by definition, the necessity is evident. Conversely, we define a relation on the vertex set V as follows:

a D b if a = b ; or there is a path from a to b .

It is not difficult to see that D is an equivalence on V . Consider the quotient V / D = { V α : α ∈ A } and for V α , construct a subquiver Q α = ( V α , E α ) of Q as follows: for a , b ∈ V α ,

there is an arrow from a to b in Q α if and only if there is an arrow from a to b in Q .

It follows that Q is a disjoint union of the quivers Q α with α ∈ A . We next prove that each Q α is a complete quiver. We consider the following two cases:

  1. If Q α has exactly one vertex, then Q α is an empty graph because it has no loops; thus, Q α is a complete quiver.

  2. If Q α has more than two vertices, then for any two vertices u , v of Q α , there is a path from u to v , and furthermore by hypothesis, there is an arrow from v to u in Q . Therefore, there is an arrow from v to u in Q α , and by definition, Q α is a complete quiver.

However, Q α is a complete quiver. Consequently, Q is a disjoint union of complete quivers.

Definition 4.2

Let Q = ( V , E ) be a quiver with vertex set V and arrow set E . Construct a quiver Q ¯ P C with vertex set V and in which for u , v ∈ V , there is an arrow from u to v in Q ¯ P C if u ≠ v and there is a path from u to v in Q . The quiver Q ¯ P C is called the path-connected quiver of Q , written Q ¯ P C .

By definition, it is easy to know that the path-connected quiver of a quiver always has no loops.

4.2 Generalized path algebras

We recall the definition of generalized path algebras.

Let I be a nonempty set and K a field. For any i , j , u , v ∈ I , A i j is a vector space over the field K , and there, exists K -linear mapping μ i j u from A i j ⊗ K A j u into A i u , written μ i j u ( x ⊗ y ) = x y , such that x ( y z ) = ( x y ) z for any x ∈ A i j , y ∈ A j u , z ∈ A u v , then the set { A i j , I , μ i j u } is a Γ -system with index I over the field K . Similar to the generalized matrix ring, we obtain a K -algebra, called a generalized matrix algebra, or a gm algebra in short, and written as GℳA ( A i j , I , μ i j u ) , or GℳA in short.

Assume that D = ( V , E ) is a quiver (possibly an infinite quiver and also not a simple graph) with vertex set V and arrow set E . Let Ω = GℳA ( Ω i j , V , μ i j u ) be a generalized matrix algebra over the field K satisfying the following conditions:

  1. Ω has a generalized matrix unit { e i i : i ∈ V } .

  2. Ω i j = 0 for any i , j ∈ V with i ≠ j .

The sequence x = a i 0 x i 0 i 1 a i 1 x i 1 i 2 ⋯ x i n − 1 i n a i n is called a generalized path, or an Ω -path, from i 0 to i n via arrows x i 0 i 1 , x i 1 i 2 , … , x i n − 1 i n , where 0 ≠ a i p ∈ Ω i p i p for p = 0 , 1 , 2 , … , n . In this case, n is called the length of x , written l ( x ) .

For two Ω -paths x = a i 0 x i 0 i 1 a i 1 x i 1 i 2 ⋯ x i m − 1 i m a i m and y = b j 0 y j 0 j 1 b j 1 y j 1 j 2 ⋯ y j n − 1 j n b j n with i m = j 0 , we define the multiplication of x and y as follows:

(4.1) x y = a i 0 x i 0 i 1 a i 1 x i 1 i 2 ⋯ x i m − 1 i m ( a i m b j 0 ) y j 0 j 1 b j 1 y j 1 j 2 ⋯ y j n − 1 j n b j n .

Denote by A i j ′ the vector space over the field K with basis consisting of all Ω -paths from i to j with length ≥ 1 . Let B i j be the subspace spanned by all elements:

(4.2) a i 0 x i 0 i 1 a i 1 x i 1 i 2 ⋯ x i k − 1 i k ∑ r = 1 n a i k ( r ) x i k + 1 i k + 2 ⋯ x i m − 1 i m a i m − ∑ r = 1 n a i 0 x i 0 i 1 a i 1 x i 1 i 2 ⋯ x i k − 1 i k a i k ( r ) x i k + 1 i k + 2 ⋯ x i m − 1 i m a i m ,

where i 0 = i , i m = j , a i k ( l ) ∈ Ω i k i k , and x i p i p + 1 is an arrow, p = 0 , 1 , … , m − 1 . Let A i j = A i j ′ / B i j when i ≠ j and A i i = ( A i i ′ + Ω i i ) / B i i , written [ α ] = α + B i j for any generalized path α from i to j . We can obtain a K -linear mapping κ i j u from A i j ⊗ K A j u to A i u induced by (4.1). We write a instead of [ a ] when a ∈ Ω . So, ( A i j , V , κ i j u } is a Γ -system. It is not difficult to know that e i i x i j = x i j = x i j e j j for any x i j from i to j . Moreover, { e i i : i ∈ V } is a generalized matrix unit of the Γ -system ( A i j , V , κ i j u ) .

The notion of generalized path algebras is originally defined in [24]. For generalized path algebras, also see [25].

Definition 4.3

The aforementioned generalized matrix algebra GℳA ( A i j , V , κ i j u ) is called the generalized path algebra of the quiver D over the generalized matrix algebra Ω , or the Ω -path algebra, written K ( D , Ω ) . If, in addition, Ω i i = K e i i for any i ∈ V , then K ( D , Ω ) is called a path algebra of the quiver D over the field K , written K ( D ) .

It is worthy to record here that for a generalized path algebra K ( D , Ω ) , by (4.2), it follows that for any nonzero elements,

x = a i 0 x i 0 i 1 a i 1 x i 1 i 2 ⋯ x i m − 1 i m a i m , y ∈ K ( D , Ω ) ,

we have

  1. x = 0 if and only if a i r = 0 for some 0 ≤ r ≤ m ;

  2. [ x ] = [ y ] in K ( D , Ω ) if and only if x = y regarded as sequences.

Let Gℳ = Gℳ ( A i j , I , μ i j l ) be a generalized matrix ring, and construct a quiver Q ( Gℳ ) with vertex set I and in which there is an arrow from i to j if and only if i ≠ j and A i j ≠ 0 . We call the quiver Q ( Gℳ ) the Γ -quiver of the generalized matrix algebra Gℳ ( A i j , I , μ i j l ) . Obviously, Q ( Gℳ ) is a quiver without loops.

We next establish the relationship between a quiver and the Γ -quiver of its generalized path algebra.

Lemma 4.4

Let D = ( V , E ) be a quiver. If K ( D , Ω ) is a generalized path algebra of D over the generalized matrix ring Ω , then

  1. Q ( K ( D , Ω ) ) is just the path connected quiver D ¯ P C of D .

  2. For any i , j , u ∈ V , if x is a nonzero element of A i j , then x A j u ≠ 0 whenever A j u ≠ 0 .

Proof

(i). Notice that Q ( K ( D , Ω ) ) and D ¯ P C have the same vertex set. So, we need only to see whether Q ( K ( D , Ω ) ) and D ¯ P C have the same arrow set. It follows from the following implications: There is an arrow from u to v in Q ( K ( D , Ω ) ) if and only if u ≠ v and A u v ≠ ∅ ; if and only if u ≠ v and A u v ′ ≠ ∅ ; if and only if there is a Ω -path a i 0 x i 0 i 1 a i 1 x i 1 i 2 ⋯ x i n − 1 i n a i n , where i 0 = u , i n = v ; if and only if there is a Ω -path e i 0 i 0 x i 0 i 1 e i 1 i 1 x i 1 i 2 ⋯ x i n − 1 i n e i n i n , where i 0 = u , i n = v ; if and only if there is a path x i 0 i 1 x i 1 i 2 ⋯ x i n − 1 i n , where i 0 = u , i n = v ; if and only if there is an edge from u to v in D ¯ P C .

(ii). Let x = a i 0 x i 0 i 1 a i 1 x i 1 i 2 ⋯ x i n − 1 i n a i n , where x i 0 i 1 , x i 1 i 2 , … , x i n − 1 i n ∈ E , i 0 = i , i n = j and 0 ≠ a i p for p = 0 , 1 , 2 , … , n − 1 . Obviously, y = e j 0 j 0 y j 0 j 1 e j 1 j 1 y j 1 j 2 ⋯ y j n − 1 j n e j n j n , where j 0 = j , j n = u and y j p j p + 1 ∈ E for p = 0 , 1 , … , n − 1 , is a nonzero element of A j u . Then,

x y = a i 0 x i 0 i 1 a i 1 x i 1 i 2 ⋯ x i n − 1 i n ( a i n e j 0 j 0 ) y j 0 j 1 e j 1 j 1 y j 1 j 2 ⋯ y j n − 1 j n e j n j n = a i 0 x i 0 i 1 a i 1 x i 1 i 2 ⋯ x i n − 1 i n a i n y j 0 j 1 e j 1 j 1 y j 1 j 2 ⋯ y j n − 1 j n e j n j n ≠ 0 ,

which results (ii).□

By Theorem 3.2 and Lemma 4.4, we may prove the following theorem.

Theorem 4.5

Let K ( D , Ω ) be a generalized path algebra. Then K ( D , Ω ) is semiprime (semiprimitive, respectively) if and only if

  1. D ¯ P C of D is a disjoint union of complete quivers;

  2. for any i ∈ V , A i i is semiprime (semiprimitive, respectively).

Proof

Suppose that K ( D , Ω ) is semiprime (semiprimitive, respectively). Theorem 3.2 immediately results (ii). Notice that A i j ≠ 0 if and only if there is a Ω -path from i to j ; if and only if there is a path from i to j in the quiver D ; if and only if there is an arrow from i to j in D ¯ P C . We can observe that if in D ¯ P C , there is a path i 1 → i 2 → ⋯ → i n , then A i k i k + 1 ≠ 0 for i = 1 , 2 , … , n − 1 , so that by Lemma 4.4, A i 1 i 2 A i 2 i 3 ⋯ A i n − 1 i n ≠ 0 . This means that there is a Ω -path from i 1 to i n . It follows that A i 1 i n ≠ 0 . By (3.3) in the proof of Theorem 3.2, this implies that A i n i 1 ≠ 0 , thereby there is an arrow from i n to i 1 in Q ( K ( D , Ω ) ) . It follows from Lemma 4.4 (i) that there is an arrow in D ¯ P C . Now by (OB2), D ¯ P C is a disjoint union of complete quivers.

Conversely, assume that given conditions hold. For a nonzero element x ∈ A i j , we have A i j ≠ 0 , so that there is a path from i to j in D ¯ P C , it follows from (OB2) that there is an arrow from j to i in D ¯ P C , thus A j i ≠ 0 . Again by Lemma 4.4, x A j i ≠ 0 and similarly, A j i x ≠ 0 . Now by Theorem 3.2, K ( D , Ω ) is semiprime (semiprimitive, respectively).□

Also, by Theorem 3.3 and Lemma 4.4, we have

Theorem 4.6

Let K ( D , Ω ) be a generalized path algebra. Then K ( D , Ω ) is prime (primitive, respectively) if and only if

  1. D ¯ P C is a complete quiver;

  2. for any i ∈ V , A i i is prime (primitive, respectively).

Proof

For the necessity, it suffices to verify that D ¯ P C is a complete quiver.

  1. If ∣ I ∣ = 1 , then Q ( K ( D , Ω ) ) is an empty graph since it has no loops, and by Lemma 4.4, D ¯ P C is a complete quiver.

  2. Assume that ∣ I ∣ ≥ 2 . For any i , j ∈ I with i ≠ j , by definition, A i i ≠ 0 and A j j ≠ 0 , and by Theorem 3.3 (i), A i i A i j A j j ≠ 0 . It follows that A i j ≠ 0 , so that there is an arrow form i to j in Q ( K ( D , Ω ) ) . Similarly, we may prove that there is an arrow form j to i in Q ( K ( D , Ω ) ) . Therefore, Q ( K ( D , Ω ) ) is a complete quiver, and by Lemma 4.4 (i), D ¯ P C is a complete quiver.

To verify the sufficiency, we consider the following two cases:

  1. If ∣ I ∣ = 1 , then Q ( K ( D , Ω ) ) is an empty graph, and K ( D , Ω ) ≅ A i i , and this means that Condition (i) in Theorem 3.3 holds since each A i i has a unity. It follows that K ( D , Ω ) is prime.

  2. Assume that ∣ I ∣ ≥ 2 . In this case, by Q ( K ( D , Ω ) ) is a complete quiver, there is an arrow from j to k in Q ( K ( D , Ω ) ) for any j , k ∈ I . This shows that A j k ≠ 0 . By Lemma 4.4, x A j k y ≠ 0 for any i , j , k , l ∈ I and nonzero elements x ∈ A i j , y ∈ A k l . Now by Theorem 3.3, K ( D , Ω ) is prime.

However, K ( D , Ω ) is prime. Similarly, we may verify the primitive case. We complete the proof.□

We may now prove the following proposition.

Proposition 4.7

Let K ( D , Ω ) be a generalized path algebra and i ∈ V .

  1. If D has no paths from i to i , then A i i is prime (primitive, semiprime and semiprimitive, respectively) if and only if so is Ω i i .

  2. If D has paths from i to i , then the following conditions are equivalent:

    1. A i i is semiprime;

    2. ann ℓ ( Ω i i ) = 0 and ann r ( Ω i i ) = 0 , where ann ℓ ( X ) ( ann r ( X ) ) is the left (right) annihilator of X ;

    3. A i i is prime.

Proof

(i). If D has no paths from i to i , then A i i ′ = 0 and so A i i = [ Ω i i ] . It follows that A i i is isomorphic to Ω i i , which results (i).

(ii). Assume that D has paths from i to i . We need only to verify that ( 1 ) ⇒ ( 2 ) and ( 2 ) ⇒ ( 3 ) since a prime ring is semiprime.

( 1 ) ⇒ ( 2 ) . Suppose that A i i is semiprime. We assume contrariwise at least one of ann ℓ ( Ω i i ) ≠ 0 and ann r ( Ω i i ) ≠ 0 holds. Without loss of generality, we let ann ℓ ( Ω i i ) ≠ 0 and u ∈ ann ℓ ( Ω i i ) , so that u Ω i i = 0 . Consider the generalized path

x = a i 0 x i 0 i 1 a i 1 x i 1 i 2 ⋯ x i m − 1 i m u

via arrows x i 0 i 1 , x i 1 i 2 , … , x i n − 1 i n with i 0 = i = i m , and for any generalized path,

y = b j 0 y j 0 j 1 b j 1 y j 1 j 2 ⋯ y j n − 1 j n b j n

via arrows y j 0 j 1 , y j 1 j 2 , … , y j n − 1 j n with j 0 = i = j n . Obviously, b j 0 ∈ Ω i i . Therefore,

x y = a i 0 x i 0 i 1 a i 1 x i 1 i 2 ⋯ x i m − 1 i m ( u b j 0 ) y j 0 j 1 b j 1 y j 1 j 2 ⋯ y j n − 1 j n b j n = a i 0 x i 0 i 1 a i 1 x i 1 i 2 ⋯ x i m − 1 i m 0 y j 0 j 1 b j 1 y j 1 j 2 ⋯ y j n − 1 j n b j n = 0 ,

and hence, x b j 0 = 0 , thereby by the arbitrariness of b j 0 , x A i i ′ = 0 . It follows that x A i i x = 0 , contrary to the hypothesis that A i i is semiprime. It results ( 2 ) .

( 2 ) ⇒ ( 3 ) . Assume that ( 2 ) is satisfied. We let contrarily w , z ∈ K ( D , Ω ) \ { 0 } such that

(4.3) w A i i z = 0 .

Let w = ∑ k = 1 r w k and z = ∑ k = 1 s z k , where w k , z k are generalized paths, and

  1. l ( w 1 ) ≥ l ( w 2 ) ≥ ⋯ ≥ l ( w r ) ;

  2. l ( z 1 ) ≥ l ( z 2 ) ≥ ⋯ ≥ l ( z s ) ;

  3. w 1 ∉ Ω i i w k Ω i i for 2 ≤ k ≤ r ;

  4. z 1 ∉ Ω i i z l Ω i i for 2 ≤ l ≤ s .

The equality (4.3) can derive that w Ω i i z = 0 , so that for any d ∈ Ω i i ,

w 1 d z 1 + w 1 d z 2 + ⋯ + w r d z s = 0 ,

and hence, by (E2) and comparing the lengths of generalized paths w i d z j , w 1 d z 1 = 0 . It follows that

(4.4) 0 = w 1 Ω i i z 1 = a i 0 x i 0 i 1 a i 1 x i 1 i 2 ⋯ x i m − 1 i m ( u Ω i i v ) y j 0 j 1 b j 1 y j 1 j 2 ⋯ y j n − 1 j n b j n ,

where w 1 = a i 0 x i 0 i 1 a i 1 x i 1 i 2 ⋯ x i m − 1 i m u and z 1 = v y j 0 j 1 b j 1 y j 1 j 2 ⋯ y j n − 1 j n b j n . Again by (E1), the equality (4.4) can imply that u Ω i i v = 0 . We have u Ω i i = 0 and Ω i i v = 0 by picking u = 1 i or v = 1 i . This is contrary to (2). Therefore, A i i is prime.□

Lemma 4.8

Let Q be a quiver and K a field. Then the following statements are true for the path algebra K ( Q ) :

  1. If Q has no paths from i to i , then A i i ≅ K .

  2. If Q has paths from i to i , then A i i is semiprimitive.

Proof

(i). If Q has no paths from i to i , then A i i = [ Ω i i ] = [ K e i i ] ≅ K e i i . But K e i i ≅ K , so A i i ≅ K .

(ii). We assume on the contrary that A i i is not semiprimitive. With notations in the proof of ( 2 ) ⇒ ( 3 ) in Proposition 4.7, assume that w is a nonzero element in r a d A i i , and further, let z be a nonzero element of A i i such that w z + w + z = 0 . Without the loss of generality, we let z = ∑ k = 1 s z k , and each z k has the same properties as ( 2 ) ⇒ ( 3 ) in Proposition 4.7. So,

(4.5) w 1 z 1 + ⋯ + w r z s + z 1 + ⋯ + z s + w 1 + ⋯ + w r = w z + w + z = 0 .

Consider that the length of w i z j is bigger than those of w i and z j , equation (4.5) derives that w 1 z 1 + ⋯ + w r z s = 0 . Notice that the length of w 1 z 1 is maximum among all w i z j , and this equation implies that w 1 z 1 = 0 . It follows that w 1 = 0 or z 1 = 0 since w 1 and z 1 are both Ω -paths from i to i . Now by the maximality of l ( w 1 ) and l ( z 1 ) , all w i are zero or all z j are zero. Therefore, w = 0 or z = 0 , contrary to that w , z are neither zero elements. Consequently, each A i i is semiprimitive.□

For a path algebra K ( Q ) , by Lemma 4.8, each Ω i i is semiprimitive and of course, semiprime. Now, the following theorem is an immediate consequence of Theorem 4.5.

Theorem 4.9

Let Q be a quiver and K a field. Then, the following conditions are equivalent:

  1. K ( Q ) is semiprime;

  2. Q ¯ P C is the disjoint union of complete quivers;

  3. K ( Q ) is semiprimitive.

By Proposition 4.7 and since a field is prime, each A i i of the path algebra K ( Q ) is prime. Theorem 4.6 results in the following theorem.

Theorem 4.10

Let Q be a quiver and K a field. Then K ( Q ) is prime if and only if Q ¯ P C is a complete quiver.

Acknowledgements

The authors would like to thank the referees for their comments and suggestions, which improved the presentation of this manuscript.

  1. Funding information: This research was supported by the National Natural Science Foundation of China (grant: 11761034).

  2. Conflict of interest: The authors state no conflict of interest.

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Received: 2021-11-11
Revised: 2022-05-26
Accepted: 2022-07-20
Published Online: 2022-10-06

© 2022 Junying Guo and Xiaojiang Guo, published by De Gruyter

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

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https://doi.org/10.1515/math-2022-0487
Keywords for this article
generalized Munn ring; generalized path algebra; semiprimitive ring; semiprime ring; generalized matrix ring
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  2. A random von Neumann theorem for uniformly distributed sequences of partitions
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  102. Delta waves of the isentropic relativistic Euler system coupled with an advection equation for Chaplygin gas
  103. Multiplicity and minimality of periodic solutions to fourth-order super-quadratic difference systems
  104. On the reciprocal sum of the fourth power of Fibonacci numbers
  105. Averaging principle for two-time-scale stochastic differential equations with correlated noise
  106. Phragmén-Lindelöf alternative results and structural stability for Brinkman fluid in porous media in a semi-infinite cylinder
  107. Study on r-truncated degenerate Stirling numbers of the second kind
  108. On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq
  109. Some new characterizations of finite p-nilpotent groups
  110. A Billingsley type theorem for Bowen topological entropy of nonautonomous dynamical systems
  111. F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface
  112. On modules related to McCoy modules
  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
  126. Dual uniformities in function spaces over uniform continuity
  127. Review Article
  128. On Hahn-Banach theorem and some of its applications
  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
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