  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
$m$.
