Ring Theory Seminar

Ehsaan Hossain, Department of Pure Mathematics, University of Waterloo

“Morita Theory X: The Main Structure Theorem”

Recall that two rings R, S are Morita equivalent if their module categories are equivalent: ModR ≡ ModS. We write R ∼ S for this. Using what we’ve built up so far, we can now show Morita’s main theorems: R S if and only if S ≃ End(PR) where PR is some progenerator in ModR. We’ll see that the endomorphism ring of a progenerator is pretty much the same thing as a full corner of Mn(R), and from this we may deduce many Morita invariant ring properties, and show that if R S then R and S have isomorphic ideal lattices.

MC 5403