Let R be a semisimple ring. (i) Prove, for all n ≥1, that Torn^R(A, B)={0} for all right R-modul

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Let R be a semisimple ring. (i) Prove, for all n ≥1, that...

Let $R$ be a semisimple ring.
(i) Prove, for all $n \geq 1$, that $\operatorname{Tor}_{n}^{R}(A, B)=\{0\}$ for all right $R$-modules $A$ and all left $R$-modules $B$.
(ii) Prove, for all $n \geq 1$, that $\operatorname{Ext}_{R}^{n}(A, B)=\{0\}$ for all left $R$-modules $A$ and $B$.

Key Concepts

Ext Functor

The Ext functor is a derived functor that measures the failure of the Hom functor to be exact, capturing information about module extensions. In the context of semisimple rings where every module is injective, higher Ext groups vanish due to the existence of trivial injective resolutions. Thus, Ext_R^{n}(A, B) = {0} for n ? 1.

Injective Modules

Injective modules are characterized by the property that every module embeds into an injective module such that certain extension problems can always be solved. In semisimple rings, every module is injective. This property ensures that using injective resolutions in the definition of derived functors like Ext yields vanishing higher Ext groups.

Tor Functor

The Tor functor is a derived functor that measures the failure of tensor products to be exact. When working over a semisimple ring, since every module is projective and projective resolutions are trivial, the higher Tor groups vanish. This is why Tor_n^{R}(A, B) = {0} for n ? 1.

Semisimple Ring

A semisimple ring is one in which every module is a direct sum of simple modules. This structure implies that every module is both projective and injective. The semisimplicity of the ring significantly simplifies homological algebra computations and ensures that many higher derived functors vanish.

Projective Modules

Projective modules are modules that have the lifting property with respect to surjective homomorphisms. In a semisimple ring, every module is projective. This means that every module has a trivial projective resolution, which in turn implies that higher Tor groups vanish when computed using these resolutions.

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