Abstract Algebra Answers

Let R be a finite-dimensional k-algebra, M be an R-module and E = EndRM. Show that if f ∈ E is such that f(M) ⊆ (rad R)M, then f ∈ rad E.

Let R be a finite-dimensional k-algebra which splits over k. Show that, for any field K ⊇ k, rad (RK) = (rad R)K.

Let M,N be finite-dimensional modules over a finite-dimensional k-algebra R. For any field K ⊇ k, show that MK and NK have a common composition factor as RK-modules iff M and N have a common composition factor as R-modules.

For any nonzero ring k and any group G, show that the group ring kG is von Neumann regular iff k is von Neumann regular, G is locally finite, and the order of any finite subgroup of G is a unit in k.

Show that statement "for any von Neumann regular ring k, any finitely generated submodule M of a projective k-module P is a direct summand of P" is equivalent to the fact that, if k is a von Neumann regular ring, then so is Mn(k) for any n ≥ 1.

For any von Neumann regular ring k, show that any finitely generated submodule M of a projective k-module P a projective k-module.

For any von Neumann regular ring k, show that any finitely generated submodule M of a projective k-module P is a direct summand of P (and hence also a projective k-module).

Show that, if G can be right ordered, then, for any domain k, A = kG has only trivial units and is a domain. Moreover, if G <> {1}, show that A is J-semisimple.

For any group G, let
Δ(G) = {g ∈ G: [G : CG(g)] < ∞}, and Δ+(G) = {g ∈ Δ(G) : g has finite order}.
Show that Δ(G)/Δ+(G) is torsion-free abelian.

For any group G, let
Δ(G) = {g ∈ G: [G : CG(g)] < ∞}, and Δ+(G) = {g ∈ Δ(G) : g has finite order}.
Show that Δ+(G) is a characteristic subgroup of G and that Δ+(G) is the union of all finite normal subgroups of G.