Proposition 4

Let A be a ring. The following properties are equivalent:

(i) The ring A is semisimple.

(ii) Every A-module is semisimple.

(iii) There exists a generating and semisimple A-module.

(iv) There exists a faithful and semisimple A-module with finitely generated countermodule.

(v) Every A-module is projective.

(vi) Every monogenous A-module is projective.

$*$For other characterizations of semisimple rings, see Proposition 6 of X, §8, n$^o4$, p. $140.*$

Let us first prove that (i) implies (ii) and (v). Suppose that the ring A is semisimple, and consider a left A-module M. By assumption, the A-module $A_s$ is semisimple, so every free A-module is semisimple. By Proposition 20 of II, §1, No. 11, p. 218, there exist a free A-module L and a surjective A-linear mapping $u$ from L to M. Let N be the kernel of $u$. Since the A-module L is semisimple, there exists a semisimple submodule $N'$ supplementary to N in L (VIII, p. 56, Theorem 1). The A-module $N'$ is projective, and $u$ induces an isomorphism from $N'$ to M. Consequently, M is semisimple and projective.

(ii) $\Rightarrow$ (iii): If every A-module is semisimple, then the A-module $A_s$ is semisimple; it is moreover generating.

(iii) $\Rightarrow$ (iv): If M is a generating A-module, then it is faithful (VIII, p. 80, Corollary of Theorem 1), and its countermodule is finitely generated (VIII, p. 99, Corollary 1).

(iv) $\Rightarrow$ (i): Let M be a faithful and semisimple A-module with finitely generated countermodule. By Lemma 4 of VIII, p. 8, there exists a natural number $m$ such that $A_s$ is isomorphic to a submodule of $M^m$. The A-module $M^m$ is semisimple, and therefore so is $A_s$.

The implication (v) $\Rightarrow$ (vi) is immediate.

(vi) $\Rightarrow$ (i): Suppose that every monogenous A-module is projective. Let $\mathfrak{a}$ be a left ideal of A. Since the A-module $A_s/\mathfrak{a}$ is projective, there exists a left ideal $\mathfrak{b}$ of A such that $A_s$ is the direct sum of $\mathfrak{a}$ and $\mathfrak{b}$ (II, §2, No. 2, p. 231, Proposition 4). Consequently, the ring A is semisimple (VIII, p. 135, Theorem 1).