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Answer on Algebra Question for sanches

Question #16871
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.
Expert's answer
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).

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