Answer to Question #25772 in Abstract Algebra for Mohammad

Question

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.