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Proceedings of the American Mathematical Society

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

 
 

 

An extension of the Noether-Deuring theorem


Author: Klaus W. Roggenkamp
Journal: Proc. Amer. Math. Soc. 31 (1972), 423-426
DOI: https://doi.org/10.1090/S0002-9939-1972-0286839-7
MathSciNet review: 0286839
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Abstract: Let $ R$ be a commutative semilocal noetherian ring, $ \Lambda $ a left noetherian $ R$-algebra and $ M,N$ finitely generated left $ \Lambda $-modules such that $ {\operatorname{End} _\Lambda }(M)$ is of finite type over $ R$. By $ \hat R$ we denote the $ (\operatorname{rad} R)$-adic completion of $ R$.

Theorem. $ M$ is $ \Lambda $-isomorphic to a direct summand of $ N$ iff $ \hat R{ \otimes _R}M$ is $ \hat R{ \otimes _R}\Lambda $-isomorphic to a direct summand of $ \hat R{ \otimes _R}N$.

This result is used to prove a generalization of the Noether-Deuring theorem. Let $ S$ be a commutative $ R$-algebra which is a faithful projective $ R$-module of finite type; then $ M$ is $ \Lambda $-isomorphic to direct summand of $ N$ iff $ S{ \otimes _R}M$ is $ S{ \otimes _R}\Lambda $-isomorphic to a direct summand of $ S{ \otimes _R}N$.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0286839-7
Keywords: Algebra, semilocal noetherian ring, finitely presented module, isomorphism, completion, Noether-Deuring theorem
Article copyright: © Copyright 1972 American Mathematical Society

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