Abstract Algebra Answers

Let R be a k-algebra where k is a field, and M,N be left Rmodules, with dimkM <∞. It is known that, for any field extension K ⊇ k, the natural map θ : (HomR(M,N))K −→ HomRK(MK,NK) is an isomorphism of K-vector spaces. Replacing the hypothesis dimkM < ∞ by “M is a finitely presented R-module,” give a basis-free proof for the fact that θ is a K-isomorphism.

If K ⊇ k is a splitting field for a finite-dimensional k-algebra R, does it follow that K is also a splitting field for any quotient algebra R of R?

Let R be a finite-dimensional k-algebra and let L ⊇ K ⊇ k be fields. Assume that L is a splitting field for R. Show that K is a splitting field for R iff, for every simple left RL-module M, there exists a (simple) left RK-module U such that UL ∼ M.

Show that for any finite-dimensional k-algebra R and any field extension K ⊇ k, (rad R)K ⊆ rad(RK).

For a finite-dimensional k-algebra R, let T(R) = rad R + [R,R], where [R,R] denotes the subgroup of R generated by ab − ba for all a, b ∈ R. Assume that k has characteristic p > 0. Show that T(R) ⊆ {a ∈ R : a^p^m ∈ [R,R] for some m ≥ 1}, with equality if k is a splitting field for R.

Let R be a finite-dimensional k-algebra which splits over k. Show that any k-subalgebra C ⊆ Z(R) also splits over k.

Let R be a left artinian ring and C be a subring in the center Z(R) of R. If R is a finite-dimensional algebra over a subfield k ⊆ C, show that rad C = C ∩ rad R.

Let R be a left artinian ring and C be a subring in the center Z(R) of R. Show that Nil C = C ∩ rad R.