Reference request for equivalent formulations of being absolutely indecomposable

Question

I would like to ask the following question.

I am searching for a reference for the following statement:

Suppose $k$ is a perfect field. Let $A$ be a (symmetric) $k$-algebra and let $M$ be a finitely generated $A$-module. Then the following assertions are equivalent.

$\bullet$ The module $M$ is absolutely indecomposable, i.e. $M$ stays indecomposable under any ground field extension.

$\bullet$ There is an isomorphism $\text{End}_A(M)/J(\text{End}_A(M)) \cong k$.

A reference to a textbook would be cool.

Answer 1

This is Theorem 30.29 in

Curtis, Charles W.; Reiner, Irving, Methods of representation theory, with applications to finite groups and orders. Vol. I, Pure and Applied Mathematics. A Wiley-Interscience Publication. New York etc.: John Wiley & Sons. XXI, 819 p. \textsterling 40.70 (1981). ZBL0469.20001.

$A$ doesn't need to be symmetric.