Answer to Question #16871 in Algebra for sanches

Question

Let k be a ring. Show that the ring A generated over k by x, y with the relations x3 = 0 and xy + yx2 = 1 is the zero ring.

Answer 1

Right multiplying xy + yx² = 1 by x in A, we get xyx = x. Left multiplying the same equation by x², we get x²yx² = x². Therefore, x² = x(xyx)x = x³ = 0, and so 1 = xy + yx² = xy. Left multiplying this by x yields x = x²y = 0 and hence 1 = xy + yx² = 0 ∈ A, proving that A = (0).