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

Authors: Robert Gilmer and William Heinzer
Journal: Proc. Amer. Math. Soc. 106 (1989), 631-636
MSC: Primary 13B99
MathSciNet review: 969521
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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.

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Keywords: Imbedding, zero-dimensional rings, products of commutative rings
Article copyright: © Copyright 1989 American Mathematical Society

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