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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


A change of rings theorem and the Artin-Rees property

Author: M. Boratyński
Journal: Proc. Amer. Math. Soc. 53 (1975), 307-310
MSC: Primary 16A60
MathSciNet review: 0401840
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Abstract: A two-sided ideal $ \mathfrak{A}$ of the ring $ R$ is said to have the left $ AR$ property if for every left ideal $ I$ and every $ k$ there exists an $ n$ such that $ {\mathfrak{A}^n} \cap I \subset {\mathfrak{A}^k}I$. Let $ R$ be a left noetherian ring and $ \mathfrak{A}$ a two-sided ideal contained in its Jacobson radical. If $ \mathfrak{A}$ has the $ \operatorname{AR} $ property then $ {\text{l}}\;{\text{gl}}\;{\text{dim}}\;R \leqslant {\text{p}}\;{\text{dim}}\;R/\mathfrak{A} + {\text{l}}\;{\text{gl dim }}R/\mathfrak{A}$, where $ {\text{p}}\;{\text{dim}}\;R/\mathfrak{A}$ denotes the (left) projective dimension of the module $ R/\mathfrak{A}$.

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PII: S 0002-9939(1975)0401840-0
Keywords: $ \operatorname{AR} $ property, projective dimension, Jacobson radical
Article copyright: © Copyright 1975 American Mathematical Society