Determinants of matrices over commutative finite principal ideal rings

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.