Proposition 13

a) A left ideal $\mathfrak{a}$ of A admits a supplement in $A_s$ if and only if there exists an idempotent $e$ in A such that $\mathfrak{a}= Ae$. The ideal $\mathfrak{a}$ then consists of the elements $x$ of A such that $x=xe$.

b) Let $e$ and $f$ be idempotents in A. We have $Ae\subset Af$ if and only if we have $ef=e$.

c) Let M be an A-module. Then M is projective and monogenous if and only if there exists an idempotent $e$ in A such that M is isomorphic to $Ae$.

The endomorphisms of the A-module $A_s$ are the right multiplications by the elements of A. The projectors in the A-module $A_s$ are therefore the mappings $x\mapsto xe$ where $e$ is an idempotent in A. Moreover, the submodules of $A_s$ are the left ideals, and such a submodule admits a supplement if and only if it is the image of a projector (II, §1, No. 9, p. 211, Proposition 14). Assertion a) follows.

The relation $Ae\subset Af$ is equivalent to $e\in Af$. By a), it is therefore equivalent to $e=ef$; assertion b) follows.

If the A-module M is monogenous, there exists a surjective A-linear mapping $u: A_s\rightarrow M$. If, moreover, M is projective, then there exists a submodule $\mathfrak{a}$ of $A_s$ supplementary to the kernel of $u$. Then $u$ induces an isomorphism from $\mathfrak{a}$ to M. Conversely, if M is isomorphic to a direct factor of $A_s$, then it is monogenous and projective. Assertion c) therefore follows from a).