Morita Equivalence - Criteria For Equivalence

Criteria For Equivalence

Equivalences can be characterized as follows: if F:R-Mod S-Mod and G:S-Mod R-Mod are additive (covariant) functors, then F and G are an equivalence if and only if there is a balanced (S,R)-bimodule P such that _SP and P_R are finitely generated projective generators and there are natural isomorphisms of the functors, and of the functors Finitely generated projective generators are also sometimes called progenerators for their module category.

For every right-exact functor F from the category of left-R modules to the category of left-S modules that commutes with direct sums, a theorem of homological algebra shows that there is a (S,R)-bimodule E such that the functor is naturally isomorphic to the functor . Since equivalences are by necessity exact and commute with direct sums, this implies that R and S are Morita equivalent if and only if there are bimodules _RM_S and _SN_R such that as (R,R) bimodules and as (S,S) bimodules. Moreover, N and M are related via an (S,R) bimodule isomorphism: .

More concretely, two rings R and S are Morita equivalent if and only if for a progenerator module P_R, which is the case if and only if

(isomorphism of rings) for some positive integer n and full idempotent e in the matrix ring M_n(R).

It is known that if R is Morita equivalent to S, then the ring Cen(R) is isomorphic to the ring Cen(S), where the Cen(-) denotes the center of the ring, and furthermore R/J(R) is Morita equivalent to S/J(S), where J(-) denotes the Jacobson radical.

While isomorphic rings are Morita equivalent, Morita equivalent rings can be nonisomorphic. An easy example is that a division ring D is Morita equivalent to all of its matrix rings M_n(D), but cannot be isomorphic when n > 1. In the special case of commutative rings, Morita equivalent rings are actually isomorphic. This follows immediately from the comment above, for if R is Morita equivalent to S, .