An Extension of the Noether-Deuring Theorem

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 $S{ \otimes _R}N$.