Answer to Question #23458 in Abstract Algebra for jeremy

Question

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

Answer 1

Assume n > 1. To see that Ris not isomorphic P, note that, the ideals of P are of theform M₂(kZ) = kM₂(Z) = kP, where k∈ Z. Now R has an ideal M₂(nZ) (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 Ris not isomorphic P.