Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Zero-dimensionality in commutative rings

Authors: Robert Gilmer and William Heinzer
Journal: Proc. Amer. Math. Soc. 115 (1992), 881-893
MSC: Primary 13C15; Secondary 13A99, 13E10
MathSciNet review: 1095223
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Abstract: If $ {\left\{ {{R_\alpha }} \right\}_{\alpha \in A}}$ is a family of zero-dimensional subrings of a commutative ring $ T$, we show that $ { \cap _{\alpha \in A}}{R_\alpha }$ is also zero-dimensional. Thus, if $ R$ is a subring of a zero-dimensional subring $ [unk]\;T$ (a condition that is satisfied if and only if a power of $ rT$ is idempotent for each $ r \in R$, then there exists a unique minimal zero-dimensional subring $ {R^0}$ of $ T$ containing $ R$. We investigate properties of $ {R^0}$ as an $ R$-algebra, and we show that $ {R^0}$ is unique, up to $ R$-isomorphism, only if $ R$ itself is zero-dimensional.

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Keywords: Zero-dimensional ring, von Neumann regular ring, minimal zero-dimensional extension ring, products of commutative rings, imbeddability in a zero-dimensional ring
Article copyright: © Copyright 1992 American Mathematical Society