Example 1

Let A be a $k$-algebra and $e$ an idempotent in A such that $AeA = A$. The $k$-algebra $eAe$ can be identified with the endomorphism $k$-algebra of the submodule $eA_d$ of $A_d$ (Lemma 4). Since $AeA = A$, it follows from Proposition 14 that $eA_d$ is an invertible $(eAe,A)_k$-bimodule, so that the $k$-algebras $eAe$ and A are Morita equivalent. Moreover, Morita’s theorem (VIII, p. 103) implies the following results:

a) Let M and N be left A-modules. Every $eAe$-linear mapping from $eM$ to $eN$ extends uniquely to an A-linear mapping from M to N.

b) Every left module over the $k$-algebra $eAe$ is isomorphic to a module of the form $eM$, where M is a left A-module.