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 $R$ in a zero-dimensional ring. The case where $R$ is a product of zero-dimensional rings is of special interest. We show (1) if the zero ideal of $R$ admits a unique representation as an irredundant intersection of (strongly primary) ideals, then $R$ need not be imbeddable in a zero-dimensional ring, and (2) for a family $\left \{ {{R_\alpha }} \right \}$ of zero-dimensional rings, $R = \prod {R_\alpha }$ is imbeddable in a zero-dimensional ring if and only if $R$ itself is zero-dimensional.