Question

For any integer n > 0, show that
R =
Z nZ
Z Z
is a prime ring.

Answer 1

To see that R is prime, viewit as a subring of P = M₂(Z). Note that nP ⊆ R. If a, b ∈ R are such that aRb = 0, then naPb ⊆ aRb = 0, and hence aPb = 0. Since P isa prime ring, we conclude that a = 0 or b = 0.