Abstract Algebra Answers

Let R be a left primitive ring such that a(ab − ba) = (ab − ba)a for all a, b ∈ R. Show that R is a division ring.

Let k be a field of characteristic zero. Represent the Weyl algebra R = k<x, y > /(xy − yx − 1) as a dense ring of linear transformations on an infinite-dimensional k-vector space V , and restate the Density Theorem in this context as a theorem on differential operators.

Let A be a subring of a field K. Show that the subring S of K <x1, . . . , xn> (n ≥ 2) consisting of polynomials with constant terms in A is a left primitive ring with center A.

For a field k, construct two left modules V, V' over the free algebra R = k <x, y> as follows. Let V = V' =(sum over i)eik. Let R act on V by: xei = ei−1, yei = ei2+1, and let R act on V ' by xei = ei−1, yei = ei2+2 (with the convention that e0 = 0). Show that V, V ' are nonisomorphic faithful simple left R-modules.

Keep the notations above and define f, g ∈ E by g(ei) = ei+1, f(ei) = ei−1 (with the convention that e0 = 0). Let R be the k-subalgebra of E generated by f and g. Show that R is isomorphic to S : =k <x, y> /(xy − 1), with a k - isomorphism matching f with x and g with y.

Keep the notations above and define f, g ∈ E by g(ei) = ei+1, f(ei) = ei−1 (with the convention that e0 = 0). Let R be the k-subalgebra of E generated by f and g. Show that R acts densely on Vk.

Let V =(infinite direct sum)eik where k is a field. For any n, let Sn be the set of endomorphisms λ ∈ E = End(Vk) such that λ stabilizes (sum over i=1,n)eik and λ(ei) = 0 for i ≥ n + 1. For any i, j, let Eij ∈ E be the linear transformation which sends ej to ei and all ej' to zero. Show that any k-subalgebra R of E containing all the Eij ’s is dense in E and hence left primitive.

Let V =(infinite direct sum)eik where k is a field. For any n, let Sn be the set of endomorphisms λ ∈ E = End(Vk) such that λ stabilizes (sum over i=1,n)eik and λ(ei) = 0 for i ≥ n + 1. Show that S =(union over n)Sn ⊆ E is a dense set of linear transformations.

Let E = End(Vk) where V is a right vector space over the division ring k., and let R ⊆ E be a dense subring. Deduce that the set S = {a ∈ R : rank(a) < ∞} is a von Neumann regular ring (possibly without identity).

Let E = End(Vk) where V is a right vector space over the division ring k., and let R ⊆ E be a dense subring. For any a ∈ R with finite rank, show that a = ara for some r ∈ R.