Prove that for any primes $p$, $q$, $p\neq q$, the ring $\mathbb{Z}_{pq}$ (the ring of integers modulo pq) is semisimple, and for $p=q$ the same ring is not semisimple.
I was told that the easiest way is to observe that it has global dimension 1, so it's hereditary, not semisimple. But I don't know how to prove this.
I'm sure it's not complicated, but it eludes my mind. Thanks in advance for any useful replies.