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 *+ *yx*^{2} = 1 by *x *in *A*, we get *xyx *= *x*. Left multiplying the same equation by *x*^{2}, we get *x*^{2}*yx*^{2} = *x*^{2}. Therefore, *x*^{2} = *x*(*xyx*)*x *= *x*^{3} = 0, and so 1 = *xy *+ *yx*^{2} = *xy*. Left multiplying this by *x *yields *x *= *x*^{2}*y *= 0 and hence 1 = *xy *+ *yx*^{2} = 0 *∈** A*, proving that *A *= (0).

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