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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.
Expert's answer
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.

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