On the imbedding of a direct product into a zero-dimensional commutative ring

Abstract

This paper addresses questions related to results of M. Arapovic concerning imbeddability of a commutative unitary ring R R in a zero-dimensional ring. The case where R R is a product of zero-dimensional rings is of special interest. We show (1) if the zero ideal of R R admits a unique representation as an irredundant intersection of (strongly primary) ideals, then R R need not be imbeddable in a zero-dimensional ring, and (2) for a family { R α } \left \{ {{R_\alpha }} \right \} of zero-dimensional rings, R = ∏ R α R = \prod {R_\alpha } is imbeddable in a zero-dimensional ring if and only if R R itself is zero-dimensional.