Answer to Question #17181 in Abstract Algebra for Melvin Henriksen

Question

Let k be a field of characteristic zero, and (aij) be an m × m skew symmetric matrix over k. Let R be the k-algebra generated by x1, . . . , xm with the relations xixj − xjxi = aij for all i, j. Show that R is a simple ring iff det(aij) <> 0. In particular, R is always nonsimple if m is odd.