If P is a finitely generated left R-module, prove that P is...

If $P$ is a finitely generated left $R$-module, prove that $P$ is projective if and only if $1_{P} \in \operatorname{im} v$, where $v: \operatorname{Hom}_{R}(P, R) \otimes_{R} P \rightarrow$ $\underset{\sim}{\operatorname{Hom}}_{R}(P, P)$ is defined, for all $x \in P$, by $f \otimes x \mapsto \widetilde{f}$, where $\widetilde{f}: y \mapsto f(y) x .$ Hint. Use a projective basis.

Key Concepts

Projective Basis

A projective basis for a module P is a set of pairs (x_i, f_i), with x_i in P and f_i in Hom_R(P,R), that satisfies the condition that any element of P can be expressed uniquely as a finite linear combination of the x_i, weighted by the corresponding f_i evaluated at that element. This concept is particularly useful to demonstrate the projectivity of modules and to explicitly construct the necessary maps, as seen in the proof involving the image of v.

Hom Functor

The Hom functor assigns to a pair of R-modules the set of R-linear maps between them. In this case, Hom_R(P, R) and Hom_R(P, P) are used to study the relationships between P, its dual module, and the endomorphism ring, providing a framework to analyze conditions under which P is projective.

Finitely Generated Module

A finitely generated module is an R-module that can be generated by a finite set of elements. This concept is important as it often ensures certain finiteness and structure conditions that allow the use of techniques such as projective bases and is critical in many algebraic proofs and constructions.

Projective Module

A projective module is an R-module that has the property of lifting module homomorphisms over surjections or, equivalently, being a direct summand of a free module. This concept is essential in module theory and homological algebra because projective modules allow the construction of projective resolutions and simplify problems involving exact sequences.

Tensor Product of Modules

The tensor product of two R-modules provides a way to 'multiply' modules together to form another module, capturing bilinear forms across the modules. In the context of the problem, the tensor product Hom_R(P, R) ?_R P is used to assemble maps that approximate or recover elements of Hom_R(P, P) through the given mapping.

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