# Answer to Question #16871 in Algebra 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*+*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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