Remark 2

Let K be a commutative field, let A be a K-algebra, and let M be an A-module. The group End$_K(M)$ is endowed with an $(A$, A)-bimodule structure defined by

for every $a, a'\in A$, every $u\in$ End$_K$(M), and every $x\in M$. We endow it with the associated B-module structure. Let $e=\sum^r_{i=1}a_i\otimes a'_i$ be an element of B satisfying the conditions in part (ii) of Proposition 5, so also relations (2) and (3). If $p\in$ End$_K(M)$ is a projector whose image N is an A-submodule of M, then $ep$ is an A-linear projector with the same image.

Indeed, the image of $ep$ is contained in N. If $x$ belongs to N, then so does $a'_ix$, so that we have $p(a'_ix) =a'_ix$ and

by formula (2). We deduce from formula (3) that $aep(x) =ep(ax)$ for every $a\in A$ and every $x\in M$, which proves that $ep$ is A-linear.