Example 3

Let A be a principal ideal domain, and let P be a system of representatives consisting of irreducible elements of A (VII, §1, No. 3, p. 3). Let M be an A-module, and, for every $\pi \in P$, let $M(\pi )$ be the set of $x\in M$ such that $\pi x= 0$. By VII, §2, No. 2, p. 9, the A-module M is semisimple if and only if it is the sum of the submodules $M(\pi )$; it is then the direct sum of these submodules. This example will be generalized further on (VIII, p. 65).

Let $A_1$ and $A_2$ be algebras over a commutative ring K. In III, §4, No. 3, p. 466, we introduced the notion of left bimodule over the algebras $A_1$ and $A_2$ and showed that this notion is equivalent to that of a left module over the ring $A_1\otimes_KA_2$. We say that M is a simple (resp. semisimple, finitely generated) bimodule if it is a simple (resp. semisimple, finitely generated) module over the ring $A_1\otimes_KA_2$.