Semisimple rings have vanishing positive Tor and Ext

If $R=0$, every unital module is zero because $m=1m=0m=0$, and all assertions follow. For $R\mathrm{\ne }0$, Wedderburn–Artin theorem for semisimple rings gives $R\cong {\prod }_{j=1}^r{M}_{n_j}(D_j)$ with division rings $D_j$. By The opposite ring $R^{\mathrm{o}\mathrm{p}}$, taking opposites and transposing matrices gives $R^{\mathrm{o}\mathrm{p}}\cong {\prod }_j{M}_{n_j}(D_j^{\mathrm{o}\mathrm{p}})$. Indeed $(BA{)}^{\mathsf{T}}=A^{\mathsf{T}}{B}^{\mathsf{T}}$ when entry multiplication on the right is reversed, and the inverse of a nonzero element of $D_j$ is also its inverse in $D_j^{\mathrm{o}\mathrm{p}}$. The converse direction of the same Wedderburn–Artin theorem therefore makes $R^{\mathrm{o}\mathrm{p}}$ semisimple.

Under The Axiom of Choice, apply Equivalent module-theoretic characterizations of semisimple rings to $R$ and, by step 1.1, to $R^{\mathrm{o}\mathrm{p}}$. Every left module over either ring is projective and every short exact sequence splits. Right $R$-modules are exactly left $R^{\mathrm{o}\mathrm{p}}$-modules with action $r\cdot n=nr$, and their homomorphisms and exact sequences are unchanged. Thus projectivity and splitting hold on both sides. The AC use in the semisimple characterization is the existence of complements in arbitrary semisimple modules.

Since every short exact sequence on either side splits, Equivalent characterizations of injective modules implies that every module on that side is injective. Equivalently, a monomorphism has a retraction onto its image, so a map out of its domain extends by composing with that retraction. This establishes the injectivity assertion, not just projectivity or left-handed vanishing.

Each module now has its identity as a length-zero projective resolution and as a length-zero injective resolution. The tensor and Hom complexes formed from these have zero terms in positive degrees, so their positive homology or cohomology vanishes by Tor from a projective resolution of the right module, Ext via a projective resolution of the first variable and Ext via an injective resolution of the second variable. For supplied length-zero resolutions this calculation uses no choice.

To obtain DC locally, given an entire relation $E$ on a nonempty set $X$ and $x_0\in X$, AC chooses $s(x)\in \{y:xEy\}$ for every $x\in X$; The recursion theorem then gives $x_{n+1}=s(x_n)$ from the prescribed $x_0$. This proves the exact DC hypothesis of The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain. The identity resolutions supply enough projectives and injectives and give resolution data canonically on all modules. Consequently the hypotheses of The balanced Tor bifunctor and The balanced Ext bifunctor hold, and their comparison interfaces identify the vanishing groups of step 4.1 with the stated balanced Tor and Ext. Apply the same argument over $R^{\mathrm{o}\mathrm{p}}$ for right-module Ext.