In the previous post we learned that it is possible to recover the center of a ring from its category of left modules (as an -enriched category). For commutative rings, this justifies the idea that it is sensible to study a ring by studying its modules (since the modules know everything about the ring).

For noncommutative rings, the situation is more interesting. Two rings are said to be **Morita equivalent** if the categories are equivalent as -enriched categories. As it turns out, there exist examples of rings which are non-isomorphic but which are Morita equivalent, so Morita equivalence is a strictly coarser equivalence relation on rings than isomorphism. However, many important properties of a ring are invariant under Morita equivalence, and studying Morita equivalence offers an interesting perspective on rings on general.

Moreover, Morita equivalence can be thought of in the context of a fascinating larger structure, the **bicategory of bimodules**, which we briefly describe.