Unit group of the ring of negacirculant matrices over finite commutative chain rings

Prarinya Naksing; Somphong Jitman

doi:10.1515/spma-2025-0035

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Unit group of the ring of negacirculant matrices over finite commutative chain rings

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Abstract

Circulant matrices form an important class of matrices that have been continuously studied due to their nice algebraic structures and wide applications. In this study, we focus specifically on negacirculant matrices, which are known as extensions of the classical circulant matrices. The algebraic structures of the rings of n × n negacirculant matrices over finite fields and over finite commutative chain rings are presented. Subsequently, the algebraic structures and enumeration of the unit groups of such matrix rings are established. Additionally, the number of non-singular n × n negacirculant matrices with prescribed determinant is given in some cases. Conjectures and open problems are proposed as well as a brief discussion in the case where the underlying ring is a finite commutative ring with identity is also presented.

Keywords: negacirculant matrices; units; group of units; determinants; enumeration; finite commutative chain rings

MSC 2010: 11C20; 15B33; 16U60

1 Introduction

Circulant matrices have been introduced in [4] and extensively been studied due to their nice algebraic structures, wide applications, and various links with other mathematical objects. Algebraic structures, properties, and applications of circulant matrices have been formally summarized in [5]. Circulant matrices have been shown to have applications in many disciplines, e.g., signal processing, image processing, networked systems, communications, and coding theory. Particularly, non-singular circulant matrices over finite fields and over finite commutative chain rings (FCCRs) are applied in constructions of various families of linear codes [1,3,5,8–10,16,21,23,24]. The algebraic structures and enumeration of the unit group of the ring of n × n circulant matrices over finite fields have been studied in [18] and [20].

In this study, we focus on negacirculant matrices known as extensions of the classical circulant matrices. Negacirculant matrices have various applications especially in constructions of linear codes with good parameters [11,12,22]. Here the algebraic structures of the rings of n × n negacirculant matrices over finite fields and over FCCRs are investigated. Particularly, the algebraic structures and enumeration of the unit groups of such rings are derived. Additionally, the number of non-singular n × n negacirculant matrices with prescribed determinant is presented in some cases. A brief discussion for a general case where the ring is a finite commutative ring with identity is also given.

The study is organized as follows. In Section 2, some preliminary results are recalled and proved. The unit groups of n × n negacirculant matrices over finite fields and over FCCRs are presented in Section 3. In Section 4, the enumeration of n × n negacirculant matrices with prescribed unit determinant over finite fields and over FCCRs is provided in some cases. Summary and discussion are given in Section 5 as well as conjectures and open problems.

2 Preliminaries

In this section, fundamental results on FCCRs and matrices are reviewed, along with an exploration of properties of square elements in such rings. These are useful in the study of the ring of negacirculant matrices and its unit group. For more details, the reader may refer to [5,6,13,14].

2.1 FCCRs

A finite commutative ring with identity 1 ≠ 0 is called an FCCR if its ideals are linearly ordered by inclusion. From [13], an FCCR is known to be a principal ideal ring with a unique maximal ideal. Let R be an FCCR and let γ be a generator of its maximal ideal. Then, the ideals in R form a chain of the form

R ⊋ γ R ⊋ γ 2 R ⊋ … ⊋ γ e − 1 R ⊋ γ e R = { 0 } ,

for some positive integer e . The smallest positive integer e such that γ e = 0 is called the nilpotency index of R . The quotient ring R ⁄ γ R ≅ F q is a finite field of order q for some prime power q , and it is called the residue field of R . By abuse of notation, write a + γ R ∈ F q as the isomorphic image of an element a ∈ R . Denote by U ( R ) the set of units in R . Clearly, U ( R ) is a multiplicative group. The following lemma is useful in the study of matrices over FCCRs.

Lemma 2.1

[13,14] Let R be an FCCR of nilpotency index e and residue field F q , for some prime power q. Let γ be a generator of the maximal ideal of R. Then, the following statements hold:

∣ γ j R ∣ = q e − j for all 0 ≤ j ≤ e .
U ( R ) = { a + b ∣ a + γ R ∈ U ( F q ) and b ∈ γ R } .
∣ U ( R ) ∣ = ( q − 1 ) q e − 1 .
For each 0 ≤ i ≤ e , R ⁄ γ i R is an FCCR of nilpotency index i and residue field F q .

Let S U ( R ) = { a ∈ U ( R ) ∣ ∃ x ∈ R such that a = x 2 } is the set of square elements in U ( R ) . It is not difficult to see that S U ( R ) = { x 2 ∣ ∃ x ∈ U ( R ) } .

The following properties of square elements in finite fields are well known [25].

Lemma 2.2

[25, Theorem 6.18] Let q be an odd prime power. Then, the following statements hold:

∣ S U ( F q ) ∣ = q − 1 2 .
− 1 ∈ S U ( F q ) if and only if q ≡ 1 ( mod 4 ) .

In the light of [6, Proposition 3], we have the following results for square units in FCCRs.

Lemma 2.3

Let q be an odd prime power. Let R be an FCCR of nilpotency index e and residue field F q . Let γ be a generator of the maximal ideal of R. Then, the following statements hold:

S U ( R ) is a multiplicative subgroup of U ( R ) .
S U ( R ) = { a + b ∣ a + γ R ∈ S U ( F q ) and b ∈ γ R } .
∣ S U ( R ) ∣ = ∣ S U ( F q ) ∣ ∣ γ R ∣ = q − 1 2 q e − 1 .

As an application of the lemmas above, we have the following results.

Lemma 2.4

Let R be an FCCR with residue field F q for some odd prime power q. If q ≡ 3 ( mod 4 ) , then

S U ( R ) = { a 2 ∣ a ∈ S U ( R ) } .

Proof

Assume that q ≡ 3 ( mod 4 ) . Let Γ : S U ( R ) → S U ( R ) be defined by Γ ( a ) = a 2 for all a ∈ S U ( R ) . Then, Γ is a multiplicative group homomorphism and { a 2 ∣ a ∈ S U ( R ) } = Γ ( S U ( R ) ) ⊆ S U ( R ) . Since q ≡ 3 ( mod 4 ) , − 1 ∉ S U ( F q ) by Lemma 2.2 (ii). It follows that − 1 ∉ S U ( R ) by Lemma 2.3 (iii). Hence, ker ( Γ ) = { 1 } , which implies that Γ is injective. Consequently, Γ is surjective and ∣ S U ( R ) ∣ = ∣ Γ ( S U ( R ) ) ∣ = ∣ { a 2 ∣ a ∈ S U ( R ) } ∣ . Therefore, we have S U ( R ) = { a 2 ∣ a ∈ S U ( R ) } as desired.□

Lemma 2.5

Let R be an FCCR with residue field F q for some odd prime power q. Let c ∈ U ( R ) and let r be a positive integer. Then, the following statements holds:

There exist a , b ∈ R such that c = a 2 + b 2 .
If q ≡ 3 ( mod 4 ) , then there exist a , b ∈ R such that c = a 2 r + b 2 r .

In addition, if c is a non-square, then a , b ∈ U ( R ) .

Proof

For simplicity, c + γ R is viewed as an isomorphic image in F q . If c = a 2 for some a ∈ R , we have c = a 2 + 0 2 . Assume that c is a non-square element in R . Then, c + γ R is a non-square element in F q . It is well-known [17, Remark 6.25] that there exist x + γ R , y + γ R ∈ F q such that c + γ R = ( x + γ R ) 2 + ( y + γ R ) 2 = ( x 2 + y 2 ) + γ R . Since c + γ R is a non-square element in F q , we have x 2 + γ R , y 2 + γ R ∈ S U ( F q ) . By Lemma 2.3 (ii), it can be concluded that x 2 , y 2 ∈ S U ( R ) and c = a 2 + b 2 for some a 2 ∈ x 2 + γ R and b 2 ∈ y 2 + γ R . This completes the proof of ( i ) .

To prove (ii), assume that q ≡ 3 ( mod 4 ) . Then, S U ( R ) = { a 2 ∣ a ∈ S U ( R ) } by Lemma 2.4. The result follows.□

2.2 Circulant and negacirculant matrices

Let n be a positive integer and let R be a finite commutative ring with identity. An n × n matrix over R is called a circulant matrix if it is of the form

Cir ( a 1 , a 2 , … , a n − 1 , a n ) ≔ a 1 a 2 a 3 … a n − 1 a n a n a 1 a 2 … a n − 2 a n − 1 a n − 1 a n a 1 … a n − 3 a n − 2 ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ a 3 a 4 a 5 … a 1 a 2 a 2 a 3 a 4 … a n a 1

for some a 1 , a 2 , a 3 … , a n − 1 , a n ∈ R . Let Cir n ( R ) denote the ring of n × n circulant matrices over R . From [5], Cir n ( R ) is isomorphic to R [ X ] ⁄ ⟨ X n − 1 ⟩ as rings via the isomorphism

Cir ( a 1 , a 2 , … , a n ) ↦ a 1 + a 2 X + a 3 X 2 + ⋯ + a n X n − 1 + ⟨ X n − 1 ⟩ .

An n × n negacirculant matrix over R is defined to be an extension of a circulant of the form

NCir ( a 1 , a 2 , … , a n − 1 , a n ) ≔ a 1 a 2 a 3 … a n − 1 a n − a n a 1 a 2 … a n − 2 a n − 1 − a n − 1 − a n a 1 … a n − 3 a n − 2 ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ − a 3 − a 4 − a 5 … a 1 a 2 − a 2 − a 3 − a 4 … − a n a 1

for some a 1 , a 2 , a 3 … , a n − 1 , a n ∈ R . In the same fashion, let NCir n ( R ) denote the set of n × n negacirculant matrices over R . Similarly, NCir n ( R ) is isomorphic to R [ X ] ⁄ ⟨ X n + 1 ⟩ as rings via the isomorphism

NCir ( a 1 , a 2 , … , a n ) ↦ a 1 + a 2 X + a 3 X 2 + ⋯ + a n X n − 1 + ⟨ X n + 1 ⟩ .

Let U ( NCir n ( R ) ) (resp., U ( Cir n ( R ) ) ) denote the group of units of NCir n ( R ) (resp., Cir n ( R ) ). For each a ∈ R , let NCir n ( R , a ) = { A ∈ NCir n ( R ) ∣ det ( A ) = a } . Consequently, U ( NCir n ( R ) ) = { A ∈ NCir n ( R ) ∣ det ( A ) ∈ U ( R ) } is a disjoint union of the form

(1) U ( NCir n ( R ) ) = ⋃ a ∈ U ( R ) NCir n ( R , a ) .

For an odd positive integer n , we have ( − X ) n − 1 = − ( X n + 1 ) , which implies that ⟨ ( − X ) n − 1 ⟩ = ⟨ X n + 1 ⟩ in R [ X ] . Hence, R [ X ] ⁄ ⟨ X n − 1 ⟩ ≅ R [ X ] ⁄ ⟨ X n + 1 ⟩ via the map f ( X ) + ⟨ X n − 1 ⟩ ↦ f ( − x ) + ⟨ X n + 1 ⟩ . Then, the next lemma follows immediately.

Lemma 2.6

Let n be a positive integer and let R be an FCCR. If n is odd, then NCir n ( R ) ≅ Cir n ( R ) as rings and U ( NCir n ( R ) ) ≅ U ( Cir n ( R ) ) as multiplicative groups.

We note that some properties of Cir n ( F q ) and U ( Cir n ( F q ) ) have been investigated in [18] and [20]. In this work, it is mainly focused on NCir n ( F q ) , NCir n ( R ) , U ( NCir n ( F q ) ) , and U ( NCir n ( R ) ) .

3 Unit group of NCir n ( R )

In this section, we focus on the ring NCir n ( R ) and its unit group, where R is an FCCR and n is a positive integer. For a special case where R = F q is a finite field of odd order q , the algebraic structures and enumeration of NCir n ( F q ) and U ( NCir n ( F q ) ) are given in terms of monic irreducible factors of X n + 1 and the quotient ring F q [ X ] ⁄ ⟨ X n + 1 ⟩ in Sections 3.1 and 3.2. General results on NCir n ( R ) and U ( NCir n ( R ) ) are presented in Section 3.3.

For co-prime positive integers i , j , denote by ord i ( j ) the multiplicative order of j modulo i . Let ϕ denote the Euler’s totient function. For an odd positive integer n ′ and an integer ν ≥ 0 , we have

(2) ∑ d ∣ n ′ ϕ ( 2 ν + 1 d ) = ∑ d ∣ n ′ ϕ ( 2 ν + 1 ) ϕ ( d ) = ϕ ( 2 ν + 1 ) ∑ d ∣ n ′ ϕ ( d ) = 2 ν n ′ .

3.1 U ( NCir n ( F q ) ) with gcd ( n , q ) = 1

In [2] and [15], the factorization of X n + 1 in F q [ X ] has been presented. For a positive integer n with gcd ( n , q ) = 1 , write n = 2 ν n ′ for some odd integer n ′ ≥ 1 and integer ν ≥ 0 . From [15, Equation 3.1], we have the following factorization:

(3) X n + 1 = X 2 ν n ′ + 1 = X 2 ν + 1 n ′ − 1 X 2 ν n ′ − 1 = ∏ d ∣ n ′ Q d 2 ν + 1 ( X ) ,

where Q d 2 ν + 1 ( X ) ≔ ∏ 1 ≤ i ≤ d 2 ν + 1 gcd ( i , d 2 ν + 1 ) = 1 ( X − ω i ) is the d 2 ν + 1 th cyclotomic polynomial and ω is a primitive d 2 ν + 1 th root of unity. By [17, Theorem 2.47 (ii)], each polynomial Q d 2 ν + 1 ( X ) is factorized into a product of ϕ ( d 2 ν + 1 ) ord d 2 ν + 1 ( q ) distinct monic irreducible polynomials of the same degree ord d 2 ν + 1 ( q ) in F q [ x ] , where ϕ is the Euler’s totient function. From (3), we therefore have

(4) X n + 1 = X 2 ν n ′ + 1 = ∏ d ∣ n ′ ∏ i = 1 ϕ ( d 2 ν + 1 ) ord d 2 ν + 1 ( q ) f d , i ( X ) ,

for some monic irreducible polynomials f d , i ( X ) of degree ord d 2 ν + 1 ( q ) . Readers please refer to [2, Equation (4)] for an explicit construction of the polynomials f d , i ( X ) .

Using standard arguments in ring theory, the algebraic structures of NCir n ( F q ) can be derived. To be self-contained, the details are presented in the following theorems.

Theorem 3.1

Let q be an odd prime power and let n be a positive integer such that gcd ( n , q ) = 1 . Write n = 2 ν n ′ for some odd positive integer n ′ and integer ν ≥ 0 . Then,

NCir n ( F q ) ≅ ∏ d ∣ n ′ ∏ i = 1 ϕ ( d 2 ν + 1 ) ord d 2 ν + 1 ( q ) F q ord d 2 ν + 1 ( q ) .

Proof

From (4), we have

NCir n ( F q ) ≅ F q [ X ] ⁄ ⟨ X 2 ν n ′ + 1 ⟩ ≅ ∏ d ∣ n ′ ∏ i = 1 ϕ ( d 2 ν + 1 ) ord d 2 ν + 1 ( q ) F q [ X ] ⁄ ⟨ f d , i ( X ) ⟩ .

Since F q [ X ] ⁄ ⟨ f d , i ( X ) ⟩ ≅ F q ord d 2 ν + 1 ( q ) for all d and i , it follows that

NCir n ( F q ) ≅ ∏ d ∣ n ′ ∏ i = 1 ϕ ( d 2 ν + 1 ) ord d 2 ν + 1 ( q ) F q ord d 2 ν + 1 ( q ) .

The proof is complete.□

Theorem 3.2

Let q be an odd prime power and let n be a positive integer such that gcd ( n , q ) = 1 . Write n = 2 ν n ′ for some odd positive integer n ′ and integer ν ≥ 0 . Then,

∣ U ( NCir n ( F q ) ) ∣ = ∏ d ∣ n ′ ( q ord d 2 ν + 1 ( q ) − 1 ) ϕ ( d 2 ν + 1 ) ord d 2 ν + 1 ( q ) = q n ∏ d ∣ n ′ ( 1 − q − ord 2 ν + 1 d ( q ) ) ϕ ( 2 ν + 1 d ) ord 2 ν + 1 d ( q ) .

Proof

From Theorem 3.1, we have

NCir n ( F q ) ≅ ∏ d ∣ n ′ ∏ i = 1 ϕ ( d 2 ν + 1 ) ord d 2 ν + 1 ( q ) F q ord d 2 ν + 1 ( q ) .

Since the unit group of a product of rings is the product of unit groups of the component rings, it follows that

U ( NCir n ( F q ) ) ≅ ∏ d ∣ n ′ ∏ i = 1 ϕ ( d 2 ν + 1 ) ord d 2 ν + 1 ( q ) U ( F q ord d 2 ν + 1 ( q ) ) = ∏ d ∣ n ′ ∏ i = 1 ϕ ( d 2 ν + 1 ) ord d 2 ν + 1 ( q ) F q ord d 2 ν + 1 ( q ) \ { 0 } .

By (2), we have

∣ U ( NCir n ( F q ) ) ∣ = ∏ d ∣ n ′ ( q ord 2 ν + 1 d ( q ) − 1 ) ϕ ( 2 ν + 1 d ) ord 2 ν + 1 d ( q ) = q ∑ d ∣ n ′ ϕ ( 2 ν + 1 d ) ∏ d ∣ n ′ ( 1 − q − ord 2 ν + 1 d ( q ) ) ϕ ( 2 ν + 1 d ) ord 2 ν + 1 d ( q ) = q n ∏ d ∣ n ′ ( 1 − q − ord 2 ν + 1 d ( q ) ) ϕ ( 2 ν + 1 d ) ord 2 ν + 1 d ( q )

as desired.□

Example 1

Let q = 3 and n = 2 2 ⋅ 5 . Then, n ′ = 5 , ν = 2 , d ∈ { 1,5 } , ord 2 2 + 1 ⋅ 1 ( 3 ) = 2 , ord 2 2 + 1 ⋅ 5 ( 3 ) = 4 , ϕ ( 2 2 + 1 ⋅ 1 ) = 4 , and ϕ ( 2 2 + 1 ⋅ 5 ) = 16 . By Theorem 3.2, we have

∣ U ( NCir 2 3 ⋅ 5 ( F 3 ) ) ∣ = ∏ d ∣ 5 ( q ord 2 2 + 1 d ( 3 ) − 1 ) ϕ ( 2 2 + 1 d ) ord 2 2 + 1 d ( 3 ) = ( 3 2 − 1 ) 4 2 ( 3 4 − 1 ) 16 4 = 2,621,440,000 .

3.2 U ( NCir n ( F q ) ) with gcd ( n , q ) ≠ 1

In this part, we focus on a general case where gcd ( n , q ) ≠ 1 . The next lemma can be verified directly and it is helpful in the description of the algebraic structures of NCir n ( F q ) and U ( NCir n ( F q ) ) .

Lemma 3.3

Let q = p t be a prime power for some prime p and positive integer t. Let μ ≥ 0 be a integer and let f ( X ) be a monic irreducible polynomial of degree ℓ over F q . Then, F q [ X ] ⁄ ⟨ ( f ( X ) ) p μ ⟩ is an FCCR of nilpotency index p μ and residue field F q ℓ .

The algebraic structures and enumerations of NCir n ( F q ) and U ( NCir n ( F q ) ) are given as follows.

Theorem 3.4

Let q = p t be a prime power for some odd prime p and positive integer t. Let n be a positive integer and write n = 2 ν p μ n ′ for some odd positive integer n ′ such that p ∤ n ′ and integers ν ≥ 0 and μ > 0 . Assume the factorization of X 2 ν n ′ + 1 as in (4). Then,

NCir n ( F q ) ≅ ∏ d ∣ n ′ ∏ i = 1 ϕ ( d 2 ν + 1 ) ord d 2 ν + 1 ( q ) R d ,

where R d = F q [ X ] ⁄ ⟨ ( f d , 1 ( X ) ) p μ ⟩ is an FCCR of nilpotency index p μ and residue field F q ord d 2 ν + 1 ( q ) .

Proof

From (4), we have

(5) X n + 1 = ( X 2 ν n ′ + 1 ) p μ = ∏ d ∣ n ′ ∏ i = 1 ϕ ( d 2 ν + 1 ) ord d 2 ν + 1 ( q ) ( f d , i ( X ) ) p μ .

By Lemma 3.3, we have R d ≔ F q [ X ] ⁄ ⟨ ( f d , 1 ( X ) ) p μ ⟩ ≅ F q [ X ] ⁄ ⟨ ( f d , i ( X ) ) p μ ⟩ is an FCCR of nilpotency index p μ and residue field F q ord d 2 ν + 1 ( q ) for all d ∣ n ′ and i = 1,2 , … , ϕ ( d 2 ν + 1 ) ord d 2 ν + 1 ( q ) . Hence, by (5),

NCir n ( F q ) ≅ F q [ X ] ⁄ ⟨ X n + 1 ⟩ ≅ ∏ d ∣ n ′ ∏ i = 1 ϕ ( d 2 ν + 1 ) ord d 2 ν + 1 ( q ) F q [ X ] ⁄ ⟨ ( f d , i ( X ) ) p μ ⟩ ≅ ∏ d ∣ n ′ ∏ i = 1 ϕ ( d 2 ν + 1 ) ord d 2 ν + 1 ( q ) R d

as desired.□

Theorem 3.5

∣ U ( NCir n ( F q ) ) ∣ = q n ∏ d ∣ n ′ ( 1 − q − ord 2 ν + 1 d ( q ) ) ϕ ( 2 ν + 1 d ) ord 2 ν + 1 d ( q ) .

Proof

From Theorem 3.4, it can be deduced that

U ( NCir n ( F q ) ) ≅ ∏ d ∣ n ′ ∏ i = 1 ϕ ( d 2 ν + 1 ) ord d 2 ν + 1 ( q ) U ( R d ) ,

where R d is an FCCR of nilpotency index p μ and residue field F q ord d 2 ν + 1 ( q ) . By Lemma 2.1, we have

∣ U ( R d ) ∣ = ( q ord d 2 ν + 1 ( q ) − 1 ) q ( p μ − 1 ) ord d 2 ν + 1 ( q ) .

By (2), it follows that

∣ U ( NCir n ( F q ) ) ∣ = ∏ d ∣ n ′ ( ( q ord d 2 ν + 1 ( q ) − 1 ) q ( p μ − 1 ) ord d 2 ν + 1 ( q ) ) ϕ ( d 2 ν + 1 ) ord d 2 ν + 1 ( q ) = ∏ d ∣ n ′ q ϕ ( 2 ν + 1 d ) ∏ d ∣ n ′ q ( p μ − 1 ) ϕ ( d 2 ν + 1 ) ∏ d ∣ n ′ ( 1 − q − ord 2 ν + 1 d ( q ) ) ϕ ( 2 ν + 1 d ) ord 2 ν + 1 d ( q ) = q 2 ν n ′ q ( p μ − 1 ) 2 ν n ′ ∏ d ∣ n ′ ( 1 − q − ord 2 ν + 1 d ( q ) ) ϕ ( 2 ν + 1 d ) ord 2 ν + 1 d ( q ) = q n ∏ d ∣ n ′ ( 1 − q − ord 2 ν + 1 d ( q ) ) ϕ ( 2 ν + 1 d ) ord 2 ν + 1 d ( q ) .

This completes the proof.□

Since ∣ NCir n ( F q ) ∣ = q n , it can be concluded that

∣ NCir n ( F q , 0 ) ∣ = ∣ NCir n ( F q ) ∣ − ∣ U ( NCir n ( F q ) ) ∣ = q n − q n ∏ d ∣ n ′ ( 1 − q − ord 2 ν + 1 d ( q ) ) ϕ ( 2 ν + 1 d ) ord 2 ν + 1 d ( q ) .

3.3 Unit group of NCir n ( R )

Results on the unit group of NCir n ( R ) are investigated, where R is an FCCR and n is a positive integer. Some general discussions over finite commutative rings with identity are presented.

The following lemma gives a link between U ( NCir n ( R ) ) and U ( NCir n ( F q ) ) .

Lemma 3.6

Let R be an FCCR of nilpotency index e and residue field F q for some odd prime power q. Let n be a positive integer. Then,

∣ U ( NCir n ( R ) ) ∣ = q ( e − 1 ) n ∣ U ( NCir n ( F q ) ) ∣ .

Proof

Let γ be a generator of the maximal ideal of R and let β : NCir n ( R ) → NCir n ( R ⁄ γ R ) ≅ NCir n ( F q ) be a ring homomorphism defined by

β ( A ) = A ¯ ,

where [ a i j ] ¯ ≔ [ a i j + γ R ] for all [ a i j ] ∈ NCir n ( R ) . Then, A ∈ ker ( β ) if and only if the entries of A are in γ R . Or equivalently, A ∈ NCir n ( γ R ) . By Lemma 2.1, ∣ γ R ∣ = q e − 1 , which implies that ∣ ker ( β ) ∣ = ∣ γ R ∣ n = q ( e − 1 ) n . By the first Isomorphism Theorem for groups, it follows that

∣ NCir n ( R ) ∣ = ∣ ker ( β ) ∣ ∣ NCir n ( R ⁄ γ R ) ∣ = q ( e − 1 ) n ∣ NCir n ( R ⁄ γ R ) ∣ .

For each A ∈ NCir n ( R ) , det ( A ) ∈ U ( R ) if and only if det ( β ( B ) ) ∈ U ( R ⁄ γ R ) . It follows that A ∈ U ( NCir n ( R ) ) if and only if β ( A ) ∈ U ( NCir n ( R ⁄ γ R ) ) . Hence, β − 1 ( B ) ⊆ U ( NCir n ( R ) ) and ∣ β − 1 ( B ) ∣ = ∣ ker ( β ) ∣ for all B ∈ U ( NCir n ( R ⁄ γ R ) ) . Therefore,

∣ U ( NCir n ( R ) ) ∣ = ∣ ker ( β ) ∣ ∣ U ( NCir n ( R ⁄ γ R ) ) ∣ = q ( e − 1 ) n ∣ U ( NCir n ( F q ) ) ∣

as desired.□

Theorem 3.7

Let R be an FCCR of nilpotency index e and residue field F q for some prime power q = p t , where p is an odd prime and t is a positive integer. Let n be a positive integer and write n = 2 ν p μ n ′ for some odd positive integer n ′ such that p ∤ n and integers ν ≥ 0 and μ ≥ 0 . Then,

∣ U ( NCir n ( R ) ) ∣ = q e n ∏ d ∣ n ′ ( 1 − q − ord 2 ν + 1 d ( q ) ) ϕ ( 2 ν + 1 d ) ord 2 ν + 1 d ( q ) .

Proof

From Theorem 3.2, we have

∣ U ( NCir n ( F q ) ) ∣ = q n ∏ d ∣ n ′ ( 1 − q − ord 2 ν + 1 d ( q ) ) ϕ ( 2 ν + 1 d ) ord 2 ν + 1 d ( q ) .

By Lemma 3.6, it follows that

∣ U ( NCir n ( R ) ) ∣ = q ( e − 1 ) n ∣ U ( NCir n ( F q ) ) ∣ = q e n ∏ d ∣ n ′ ( 1 − q − ord 2 ν + 1 d ( q ) ) ϕ ( 2 ν + 1 d ) ord 2 ν + 1 d ( q ) .

The proof is complete.□

A finite commutative ring R with identity 1 ≠ 0 is called a finite commutative principal ideal ring (FCPIR) if every ideal in R is principal. From [7], we note that an FCPIR is isomorphic to a product of FCCRs. Let R ≅ ∏ i = 1 m R i be an FCPIR, where R i is an FCCR for all integers 1 ≤ i ≤ m . For each 1 ≤ i ≤ m , let ψ i : R → R i be a surjective ring homomorphism defined by

ψ i ( ( r 1 , r 2 , … , r m ) ) = r i .

Then, the map Ψ : NCir n ( R ) → NCir n ( R 1 ) × NCir n ( R 2 ) × … × NCir n ( R m ) defined by

(6) [ a i j ] ↦ ( [ ψ 1 ( a i j ) ] , [ ψ 2 ( a i j ) ] , … , [ ψ m ( a i j ) ] )

is a ring isomorphism. Based on the number ∣ U ( NCir n ( R i ) ) ∣ given in Theorem 3.7, the explicit formula of ∣ U ( NCir n ( R ) ) ∣ follows directly.

Proposition 3.8

Let n be a positive integer and let R ≅ ∏ i = 1 m R i be an FCPIR, where R i is an FCCR for all integers 1 ≤ i ≤ m . Then,

∣ U ( NCir n ( R ) ) ∣ = ∣ U ( NCir n ( R 1 ) ) ∣ ∣ U ( NCir n ( R 2 ) ) ∣ … ∣ U ( NCir n ( R m ) ) ∣ .

From the proof of Lemma 3.6, the key idea is that R is a finite commutative ring with unique maximal γ R . Using similar proof arguments, we have the following proposition.

Proposition 3.9

Let n be a positive integer and let ℛ be a finite commutative local ring with maximal ideal ℳ . Then, ℛ ⁄ ℳ ≅ F q for some prime power q and ∣ U ( NCir n ( ℛ ) ) ∣ = ∣ ℳ ∣ n ∣ U ( NCir n ( F q ) ) ∣ .

From [19, Chapter VI], we have the fact that a finite commutative ring with identity is isomorphic to a product of finite commutative local rings. Let R ≅ ∏ i = 1 m ℛ i be a finite commutative ring with identity, where ℛ i is a finite commutative local ring for all integers 1 ≤ i ≤ m . For each 1 ≤ i ≤ m , let φ i : R → ℛ i be a surjective ring homomorphism defined by

φ i ( ( r 1 , r 2 , … , r m ) ) = r i .

Then, the map Φ : NCir n ( R ) → NCir n ( ℛ 1 ) × NCir n ( ℛ 2 ) × … × NCir n ( ℛ m ) defined by

(7) [ a i j ] ↦ ( [ φ 1 ( a i j ) ] , [ φ 2 ( a i j ) ] , … , [ φ m ( a i j ) ] )

is a ring isomorphism. The following proposition can be derived immediately.

Proposition 3.10

Let n be a positive integer and let R ≅ ∏ i = 1 m ℛ i be a finite commutative ring with identity, where ℛ i is a finite commutative local ring for all integers 1 ≤ i ≤ m . Then,

∣ U ( NCir n ( R ) ) ∣ = ∣ U ( NCir n ( ℛ 1 ) ) ∣ ∣ U ( NCir n ( ℛ 2 ) ) ∣ … ∣ U ( NCir n ( ℛ m ) ) ∣ .

In general, the explicit formulas in Propositions 3.9 and 3.10 are still open.

4 Matrices with prescribed determinant in U ( NCir n ( R ) )

In this section, we focus on the number of matrices with prescribed determinant in U ( NCir n ( R ) ) , where R is an FCCR with residue field F q such that gcd ( n , q ) = 1 . A general set up where the ring is an FCPIR and an arbitrary finite commutative ring with identity is discussed at the end of this section.

Lemma 4.1

Let R be an FCCR and let n = 2 ν n ′ be a positive integer such that n ′ is odd and ν ≥ 0 is an integer. Let a , b ∈ R and let

A ν = NCir ( x ) ,

where x = ( x 1 , x 2 , … , x n ) and

x i = a if i = 1 , b if i = t ⋅ 2 ν + 1 , w h e r e t i s o d d i n { 1,2 , … , n ′ − 1 } , − b if i = t ⋅ 2 ν + 1 , w h e r e t i s e v e n i n { 1,2 , … , n ′ − 1 } , 0 otherwise .

Then,

det ( A ν ) = ( ( a − ( n ′ − 1 ) b ) ( a + b ) n ′ − 1 ) 2 ν .

Proof

For ν = 0 , we have

A 0 = NCir ( a , b , − b , … , b , − b ) = a b − b … b − b b a b … − b b − b b a … b − b ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ b − b b … a b − b b − b … b a .

For convenience, for each i ∈ { 1 , 2 , … , n } , denote by row i ( A 0 ) and col i ( A 0 ) the i th row of A 0 and the i th column of A 0 , respectively. Applying the elementary row operations row i ( A 0 ) + ( − 1 ) i row 1 ( A 0 ) → row i ( A 0 ) for all i ∈ { 2 , 3 , … , n } , we have

A 0 ∼ A ′ = a b − b … b − b a + b a + b − ( a + b ) a + b ⋮ ⋱ a + b a + b − ( a + b ) a + b .

Applying the elementary column operations col 1 ( A ′ ) − ( − 1 ) i col i ( A ′ ) → col 1 ( A ′ ) for all i ∈ { 2 , 3 , … , n } , we have

A 0 ∼ A ″ = a − ( n − 1 ) b b − b … b − b a + b a + b ⋱ a + b a + b .

Then, det ( A 0 ) = det ( A ″ ) = ( a − ( n ′ − 1 ) b ) ( a + b ) n ′ − 1 .

Assume that ν ≥ 1 . Using even times of suitable row and column swaps, we have

A ν ∼ A 0 A 0 ⋱ A 0

and det ( A ν ) = det ( A 0 ) 2 ν = ( ( a − ( n ′ − 1 ) b ) ( a + b ) n ′ − 1 ) 2 ν .□

Lemma 4.2

Let R be an FCCR with residue field F q and let n = 2 ν n ′ be a positive integer such that n ′ is odd and ν ≥ 0 is an integer. Let a ∈ U ( R ) , b ∈ R and let

B = NCir ( x ) ,

where x = ( x 1 , x 2 , … , x n ) and

x i = a if i = 1 , − b if i = n ′ + 1 , 0 otherwise .

Then,

det ( B ) = ( a 2 ν + b 2 ν ) n ′ .

Proof

Applying an elementary column operation b a − 1 col i ( B ) + col i + n ′ ( B ) → col i + n ′ ( B ) for all i ∈ { 1,2 , … , ( 2 ν − 1 ) n ′ } , it follows that B is equivalent to a lower triangular matrix whose main diagonal entries are 2 ν − 1 n ′ copies of a and n ′ copies of a + b 2 ν a 2 ν − 1 . Hence,

det ( B ) = a 2 ν − 1 n ′ a + b 2 ν a 2 ν − 1 n ′ = ( a 2 ν + b 2 ν ) n ′

as desired.□

Theorem 4.3

Let R be an FCCR with residue field F q for some odd prime power q. Let n ∈ { n ′ , 2 n ′ } be such that n ′ is an odd positive integer. If gcd ( n , q ) = 1 , then ∣ NCir n ( R , c ) ∣ > 0 for all c ∈ U ( R ) .

Proof

Let c ∈ U ( R ) . Assume that gcd ( n , q ) = 1 .

Case 1: n = n ′ . Let a = ( c − 1 + n ) n − 1 and b = ( 1 − c ) n − 1 . Then, a + b = ( c − 1 + n ) n − 1 + ( 1 − c ) n − 1 = ( c − 1 + n + 1 − c ) n − 1 = 1 and a − ( n − 1 ) b = ( c − 1 + n ) n − 1 − ( n − 1 ) ( 1 − c ) n − 1 = ( c − 1 + n − n + n c + 1 − c ) n − 1 = c . From Lemma 4.1, it follows that

det ( A 0 ) = ( a − ( n − 1 ) b ) ( a + b ) n − 1 = ( c ) ( 1 ) n − 1 = c ,

which implies that ∣ NCir n ( R , c ) ∣ > 0 .

Case 2: n = 2 n ′ . From Case 1 and Lemma 4.1, we have that det ( A 2 ) = det ( A 0 ) 2 = c 2 , which implies that ∣ NCir n ( R , α ) ∣ > 0 for all α ∈ S U ( R ) . By Lemmas 2.5 and 4.2, there exists A ∈ U ( NCir n ( R ) ) such that det ( A ) ∈ U ( R ) \ S U ( R ) . By Lemma 2.3, S U ( R ) is a subgroup of U ( R ) of index 2. Since the determinant is a multiplicative homomorphism, ∣ NCir n ( R , α ) ∣ > 0 for all α ∈ U ( R ) .□

Theorem 4.4

Let R be an FCCR of nilpotency index e and residue field F q . Let n be a positive integer written as n = 2 ν n ′ for some odd positive integer n ′ such that gcd ( n ′ , q ) = 1 and integer ν ≥ 0 . If q ≡ 3 ( mod 4 ) , then ∣ NCir n ( R , c ) ∣ > 0 for all c ∈ U ( R ) .

Proof

Assume that q ≡ 3 ( mod 4 ) . By Lemma 2.4, it follows that S U ( R ) = { x 2 ν ∣ x ∈ S U ( R ) } . By the setting in the proof of Theorem 4.3, we have that det ( A ν ) = det ( A 0 ) 2 ν = c 2 ν , which implies that ∣ NCir n ( R , α ) ∣ > 0 for all α ∈ S U ( R ) .

By Lemmas 2.5 and 4.2, there exists A ∈ U ( NCir n ( R ) ) such that det ( A ) ∈ U ( R ) \ S U ( R ) . By Lemma 2.3, S U ( R ) is a subgroup of U ( R ) of index 2. Since the determinant is a multiplicative homomorphism, ∣ NCir n ( R , α ) ∣ > 0 for all α ∈ U ( R ) .□

Corollary 4.5

Let R be an FCCR of nilpotency index e and residue field F q . Let n be a positive integer written as n = 2 ν n ′ for some odd positive integer n ′ such that gcd ( n ′ , q ) = 1 and integer ν ≥ 0 . If q ≡ 3 ( mod 4 ) or 0 ≤ ν ≤ 1 , then ∣ NCir n ( R , c ) ∣ = ∣ NCir n ( R , 1 ) ∣ for all c ∈ U ( R ) .

Proof

Let c ∈ U ( R ) . Assume that q ≡ 3 ( mod 4 ) or 0 ≤ ν ≤ 1 . By Theorems 4.3 and 4.4, we have NCir n ( R , c ) ≠ ∅ . Let C ∈ NCir n ( R , c ) and let φ : NCir n ( R , 1 ) → NCir n ( R , c ) be defined by

A ↦ C A

for all A ∈ NCir n ( R , 1 ) . It is not difficult to see that φ is a bijection. Hence, ∣ NCir n ( R , c ) ∣ = ∣ NCir n ( R , 1 ) ∣ .□

Corollary 4.6

Let R be an FCCR of nilpotency index e and residue field F q . Let n be a positive integer written as n = 2 ν n ′ for some odd positive integer n ′ such that gcd ( n ′ , q ) = 1 that and integer ν ≥ 0 . If q ≡ 3 ( mod 4 ) or 0 ≤ ν ≤ 1 , then

∣ NCir n ( R , c ) ∣ = q e n − e + 1 q − 1 ∏ d ∣ n ′ ( 1 − q − ord 2 ν + 1 d ( q ) ) ϕ ( 2 ν + 1 d ) ord 2 ν + 1 d ( q )

for all c ∈ U ( R ) .

Proof

Let c ∈ U ( R ) . From (1), U ( NCir n ( R ) ) is a disjoint union of the form

(8) U ( NCir n ( R ) ) = ⋃ a ∈ U ( R ) NCir n ( R , a ) .

Then,

∣ U ( NCir n ( R ) ) ∣ = ∑ a ∈ U ( R ) ∣ NCir n ( R , a ) ∣ = ∑ a ∈ U ( R ) ∣ NCir n ( R , 1 ) ∣ = ∣ U ( R ) ∣ ∣ NCir n ( R , 1 ) ∣

by Corollary 4.5. By Lemma 2.1 and Theorem 3.7, it can be deduced that

∣ NCir n ( R , c ) ∣ = ∣ NCir n ( R , 1 ) ∣ = ∣ U ( NCir n ( R ) ) ∣ ∣ U ( R ) ∣ = q e n ∏ d ∣ n ′ ( 1 − q − ord 2 ν + 1 d ( q ) ) ϕ ( 2 ν + 1 d ) ord 2 ν + 1 d ( q ) ( q − 1 ) q e − 1 = q e n − e + 1 q − 1 ∏ d ∣ n ′ ( 1 − q − ord 2 ν + 1 d ( q ) ) ϕ ( 2 ν + 1 d ) ord 2 ν + 1 d ( q ) .

This completes the proof.□

In general, the decomposition for the number of units in NCir n ( R ) with prescribed determinant can be given using (6) and (7).

Based on the ring isomorphism defined in (6), we have the following theorem.

Theorem 4.7

Let n be a positive integer and let R ≅ ∏ i = 1 m R i be an FCPIR, where R i is an FCCR for all integers 1 ≤ i ≤ m . Let Ψ be defined in (6). Then,

∣ NCir n ( R , r ) = ∣ NCir n ( R 1 , ψ 1 ( r ) ) ∣ ∣ NCir n ( R 2 , ψ 2 ( r ) ) ∣ … ∣ NCir n ( R m , ψ m ( r ) ) ∣

for all r ∈ U ( R ) .

Using the ring isomorphism defined in (7), the next theorem follows.

Theorem 4.8

Let n be a positive integer and R ≅ ∏ i = 1 m ℛ i be a finite commutative ring with identity, where ℛ i is a finite commutative local ring for all integers 1 ≤ i ≤ m . Let Φ be defined in (7). Then,

∣ NCir n ( R , r ) = ∣ NCir n ( ℛ 1 , φ 1 ( r ) ) ∣ ∣ NCir n ( ℛ 2 , φ 2 ( r ) ) ∣ … ∣ NCir n ( ℛ m , φ m ( r ) ) ∣

for all r ∈ U ( R ) .

While the explicit number in Theorem 4.7 can be determined in some cases based on Corollary 4.6, the explicit number in Theorem 4.8 is still open.

5 Conclusion and remarks

The algebraic structures of the rings of n × n negacirculant matrices over finite fields and over FCCRs have been presented. The characterization and enumeration of their unit groups have been completely determined. For an FCCR R with residue field F q , the number of non-singular n × n negacirculant matrices with prescribed determinant is presented in the case where gcd ( n , q ) = 1 with q ≡ 3 ( mod 4 ) or n ∈ { n ′ , 2 n ′ } for some odd integer n ′ in Corollary 4.6. Algebraic properties of the unit group of NCir n ( R ) in the case where R ring is a finite commutative ring with identity have been briefly discussed.

Based on our computation, the results in Corollaries 4.5 and 4.6 hold true without restrictions on q and n . We therefore propose the following conjectures.

Conjecture 1

Let R be a finite commutative ring with identity and let n be a positive integer. Then,

∣ NCir n ( R , 1 ) ∣ = ∣ NCir n ( R , c ) ∣ > 0

for all c ∈ U ( R ) .

Conjecture 2

∣ NCir n ( R , c ) ∣ = q e n − e + 1 q − 1 ∏ d ∣ n ′ ( 1 − q − ord 2 ν + 1 d ( q ) ) ϕ ( 2 ν + 1 d ) ord 2 ν + 1 d ( q )

for all c ∈ U ( R ) .

Assuming that Conjecture 2 holds true, the explicit formula of the one in Theorem 4.7 can be obtained by Corollary 4.6.

In general, it would be interesting to study the unit group of the ring of twistulant matrices over FCCRs or other finite commutative rings. Alternatively, the determination of the number of singular n × n twistulant matrices with prescribed determinant is an interesting problem as well.

Acknowledgements

The authors would like to thank the anonymous referees for their helpful comments.

Funding information: S. Jitman was funded by the National Research Council of Thailand and Silpakorn University under Research Grant N42A650381. P. Naksing was supported by Faculty of Science, Silpakorn University under Grant SCSU-STA-2564-03.
Author contributions: The authors have contributed equally.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Received: 2024-09-26

Revised: 2025-03-04

Accepted: 2025-03-05

Published Online: 2025-04-12

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https://doi.org/10.1515/spma-2025-0035

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negacirculant matrices; units; group of units; determinants; enumeration; finite commutative chain rings

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