Hi, i would like to know if weather or not a Galois extension of a commutative semi-local ring is also a semilocal ring.
Let $A \to B$ be a map of rings such that $B$ is a finitely generated $A$-module. Suppose $A$ is semi-local. Then $B$ is also semi-local. To see this, note that every simple $B$-module is finitely generated over $A$ hence killed by the Jacobson radical $J(A)$ of $A$ by Nakayama's Lemma. So $J(A) \subseteq J(B)$ which means that $B / J(B)$ is finitely generated as a module over $A/J(A)$ and hence has finite length as an $A$-module since $A$ is semi-local. In particular it has finite length as a $B$-module and so $B$ is semi-local.
This argument works even if the rings are non-commutative.
Thanks a lot Doctor Andarkov. By the way, i just find an article where the autor proves that given a semi-local ring $R,$ every left $R$-module is semi-local.
The article is On semilocal modules and rings Christian Lomp COMMUNICATIONS IN ALGEBRA, 27(4), 1921-1935 (1999)
The result is in Theorem 3.5