Equivalent module-theoretic characterizations of semisimple rings

Given: The hypotheses and objects in the Statement.

A unital ring $R$ is semisimple when its left regular module ${}_{R}R$ is semisimple. This is a left-module definition and uses no Jacobson radical. For the zero ring, the regular module is zero and hence semisimple; the Wedderburn-Artin theorem below is stated for nonzero rings. (A semisimple ring as a ring whose left regular module is semisimple).

Assuming the Axiom of Choice, every submodule and every quotient of a semisimple module is semisimple. (Under Choice, submodules and quotients of semisimple modules are semisimple).

For every left $R$-module $M$, the free module $R^{(M)}$ on its underlying set admits a canonical surjection ${\epsilon }_M:R^{(M)}\to M$, determined by ${\epsilon }_M(e_m)=m$. Consequently $M\cong R^{(M)}\mathrm{/}\ker {\epsilon }_M$. (Every module is a quotient of a free module).

In a short exact sequence $0\to A\stackrel{i}{\to }B\stackrel{p}{\to }C\to 0,$ a section of $p$ is a homomorphism $s:C\to B$ with $p\circ s={\mathrm{id}}_C$, and a retraction of $i$ is a homomorphism $r:B\to A$ with $r\circ i={\mathrm{id}}_A$. (Split short exact sequences, sections, and retractions).

For a short exact sequence $0\to A\stackrel{i}{\to }B\stackrel{p}{\to }C\to 0,$ the following are equivalent: 1. $p$ has a section $s:C\to B$; 2. $i$ has a retraction $r:B\to A$; 3. there is an isomorphism $\mathrm{\Phi }:A\oplus C\to B$ with $\mathrm{\Phi }(a,0)=i(a)$ and $p(\mathrm{\Phi }(a,c))=c$. (The splitting lemma for short exact sequences of modules).

A left $R$-module $P$ is projective if it has the lifting property for epimorphisms: whenever $q:E\to M$ is a surjective module homomorphism and $f:P\to M$ is a module homomorphism, there exists a module homomorphism $\tilde{f}:P\to E$ such that $q\circ \tilde{f}=f$ (def-module-homomorphism-kernel-image-and-cokernel, def-injection-surjection-bijection). (Projective modules and the lifting property).

For a left $R$-module $P$, assertions 1 to 3 below are equivalent without choice. Under the Axiom of Choice, they are also equivalent to assertion 4: 1. $P$ is projective; 2. every short exact sequence $0\to K\to E\to P\to 0$ splits; 3. ${\mathrm{Hom}}_R(P,-)$ takes every short exact sequence to a short exact sequence; 4. $P$ is a direct summand of a free module. (Equivalent characterizations of projective modules).

Assuming the Axiom of Choice, a module is semisimple if and only if every submodule has a complementary submodule. (Equivalent characterizations of semisimple modules).

Assume The Axiom of Choice. Its actual use here is the AC-qualified semisimple complement and submodule/quotient results [L8] and [L2], whose proofs use Zorn's lemma. The canonical free cover [L3] and the splitting/projectivity equivalence in clauses 1–2 of [L7] are choice-free.