Let R be a commutative ring. Show that if P is a finitely generated projective R-module, then P^{*} = Hom_{R}(P, R) is also a finitely generated projective R-module.
Let R be a domain, and for a prime ideal P in R let R_{P} denote the localization of R at P. For an R-module M let M_{P} denote the base extension M \otimes_{R} R_{P} of M to R_{P}.
Show that M_{P} is isomorphic (as an R_{P}-module) to the module of pseudo fractions m/s where m is in M and s is an element of R - P subject to the provision that the pseudo fractions m_{1}/s_{1} and m_{2}/s_{2} are equal if and only if there is an element s of R - P such that s s_{1} m_{2} = s s_{2} m_{1}.
Show that M -----> M_{P} , considered as a functor from (R-modules) to (R_{P}-modules) is an exact functor.
Let R be a (commutative) local ring. Let M and N be finitely generated R-modules for which there is an isomorphism M \otimes_{R} N ~= R . Show that M and N must be free R-modules.
If V is a finite-dimensional vector space over a field F, F[t] is the polynomial ring in one variable over F, and phi is an F-linear endomorphism of V, then the map F[t] \times V -----> V given by (f, v) ---> (f(phi))(v) defines the structure of an F[t]-module structure M(phi) on V.
Explain in terms of ``linear algebra over a field'' what it means for the F[t]-modules M(phi_{1}) and M(phi_{2}) associated with two linear endomorphisms phi_{1} and phi_{2} of V to be isomorphic.
Determine the isomorphism class of M(phi) (as an F[t]]-module) when V = F^{2} and phi(x, y) = (-y, x).
Let R be a ring and P a left R-module. Show that the following two statements are equivalent:
P is a finitely generated projective left R-module.
There exist x_{1}, ..., x_{n} \in P and f_{1}, ..., f_{n} \in Hom_{R}(P, R) such that the relation
holds for all x \in P.