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Imbeddability of a commutative ring in a finite-dimensional ring. (English) Zbl 0857.13012

Summary: In the category of commutative unitary rings we prove that, for each positive integer \(n\), there exists an \(n\)-dimensional ring \(R_n\) that is not a subring of a ring of dimension less than \(n\). We also prove existence of a ring with a unique maximal ideal that is not a subring of a finite-dimensional ring. A case of the imbeddability problem of particular interest is that for \(R= \prod R_\alpha\), where each \(R_\alpha\) is zero-dimensional. It is known that \(R\) is either zero-dimensional or infinite-dimensional. In the case where each \(R_\alpha\) is a PIR and \(R\) is infinitely-dimensional, we show that neither \(R\) nor \(R/I\), where \(I\) is the direct sum ideal of \(R\), is imbeddable in a finite-dimensional ring; moreover, \(R\) admits a DVR as a homomorphic image.

MSC:

13C15 Dimension theory, depth, related commutative rings (catenary, etc.)
13B02 Extension theory of commutative rings
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References:

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