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arXiv · 1605.06826

Determinants of Matrices over Commutative Finite Principal Ideal Rings

Parinyawat Choosuwan·Somphong Jitman·Patanee Udomkavanich
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Abstract

In this paper, the determinants of $n\times n$ matrices over commutative finite chain rings and over commutative finite principal ideal rings are studied. The number of $n\times n$ matrices over a commutative finite chain ring ${R}$ of a fixed determinant $a$ is determined for all $a\in {R}$ 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 $n\times n$ 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 $m$.

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Parinyawat Choosuwan, Somphong Jitman, Patanee Udomkavanich. 2016-05-22. Determinants of Matrices over Commutative Finite Principal Ideal Rings. https://arxiv.org/abs/1605.06826

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