I'm reading through Lang's Algebra. Lang defines a simple ring as a semisimple ring that has only one isomorphism class of simple left ideals.
On the other side, Wikipedia says that a simple ring is a non-zero ring that has no two-sided ideals except zero ideal and itself. I'm wondering, are these two definitions equivalent?
Lang shows that his definition is stronger than the one on Wikipedia, but starts Rieffel's theorem as:
Let $R$ be a ring without two-sided ideals except $0$ and $R$...
which raises suspicion about the equivalence.
1) For Lang, ring has to have the multiplicative identity.
2) Lang defines semisimple rings as rings who are semisimple as (left) modules over itself, but later shows that left-right distinction doesn't matter.