A note on subdirectly irreducible rings

Let R be a commutative subdirectly irreducible ring, with minimal ideal M. It is shown that either R is a field, or M² = 0. A construction is given which yields commutative sub-directly irreducible rings possessing nonzero-divisors, and nonzero nilpotent elements either with a unity element, or without. Such a ring without unity has been constructed by Divinsky. The same technique enables the construction of subdirectly irreducible rings with mixed additive groups.