Question #23458

Show that R =
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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