Definition 2

An A-module M is called indecomposable if it is not the direct sum of a family of submodules distinct from 0 and M.

By the corollary of Proposition 12 of II, §1, No. 8, p. 209, the following properties are equivalent:

a) The A-module M is indecomposable.

b) The A-module M is nonzero, and every direct factor submodule of M is equal to 0 or M.

c) The A-module M is nonzero, and the ring End$_A(M)$ contains no idempotent disctinct from 0 and $1_M$.

In particular, since the endomorphism ring of the A-module $A_s$ is isomorphic to the opposite ring of A, we see that the A-module $A_s$ is indecomposable if and only if the ring A is nonzero and its only idempotents are 0 and 1.