# Answer to Question #23458 in Abstract Algebra for jeremy

Question #23458

Show that R =

Z nZ

Z Z

is not isomorphic to the prime ring P = M2(Z) if n > 1.

Z nZ

Z Z

is not isomorphic to the prime ring P = M2(Z) if n > 1.

Expert's answer

Assume

*n >*1. To see that*R*is not isomorphic*P*, note that, the ideals of*P*are of theform M_{2}(*k*Z) =*k*M_{2}(Z) =*kP,*where*k**∈**Z. Now**R*has an ideal M_{2}(*n*Z) (which is, infact, an ideal of the larger ring*P*). Since this ideal of*R*isobviously not of the form*kR*for any integer*k*, it follows that*R*is not isomorphic*P*.
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