Determinants of matrices over commutative finite principal ideal rings

Abstract

In this paper, the determinants of n × n matrices over commutative finite chain rings and over commutative finite principal ideal rings are studied. The number of n × n matrices over a commutative finite chain ring R of a fixed determinant a is determined for all a ∈ R and positive integers n. Using the fact that every commutative finite principal ideal ring is a product of commutative finite chain rings, the number of n × n matrices of a fixed determinant over a commutative finite principal ideal ring is shown to be multiplicative, and hence, it can be determined. These results generalize the case of matrices over the ring of integers modulo m.

Determinants of some special matrices over commutative finite chain rings

“…Determinants of n × n matrices over CFCRs R with residue eld Fq have been established in [5]. Here, determinants of n × n diagonal and circulant matrices over R which are subrings of the matrices in [5] have been…”

“…An n × n matrix A over R is called a circulant matrix if A is of the form [6] and [14], the eigenvalues of A are of the form In this paper, we focus on the numbers cn(R, a) and dn(R, a) in the case where R is CFCRs and a ∈ R which are established in Section 3 and Section 4, respectively. While some techniques for n × n matrices from [5] are applied, the detailed proofs and counting are slightly di erent. For completeness, the full proofs and counting are given.…”

“…For completeness, the full proofs and counting are given. The reader may refer to [5] for comparison.…”

“…The number of n × n singular (resp., nonsingular) matrices over a nite eld has been given in [15]. The number of n × n matrices over commutative nite chain rings (CFCRs) of a xed determinant has been completely determined in [5]. Diagonal and circulant matrices are two interesting subfamilies of the ones in [5].…”

“…The number of n × n matrices over commutative nite chain rings (CFCRs) of a xed determinant has been completely determined in [5]. Diagonal and circulant matrices are two interesting subfamilies of the ones in [5]. Therefore, it is of natural interest to study the determinants of such matrices over CFCRs.…”