  A well-known lower bound (over finite fields and some special finite
commutative rings) on the Hamming distance of a matrix-product code (MPC) is
shown to remain valid over any commutative ring $R$. A sufficient condition is
given, as well, for such a bound to be sharp. It is also shown that an MPC is
free when its input codes are all free, in which case a generating matrix is
given. If $R$ is finite, a sufficient condition is provided for the dual of an
MPC to be an MPC, a generating matrix for such a dual is given, and
characterizations of LCD, self-dual, and self-orthogonal MPCs are presented.
Finally, results of this paper are used along with previous results of the
authors to construct novel MPCs arising from $(\sigma, \delta)$-codes. Some
properties of such constructions are also studied.
