An extension of the Noether-Deuring theorem
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- by Klaus W. Roggenkamp PDF
- Proc. Amer. Math. Soc. 31 (1972), 423-426 Request permission
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 $S{ \otimes _R}N$.References
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Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 31 (1972), 423-426
- DOI: https://doi.org/10.1090/S0002-9939-1972-0286839-7
- MathSciNet review: 0286839