---
tag: RD5WKA1VK7
kind: concept
name: Morita equivalence
msc: [16D90]
sources:
  - url: https://en.wikipedia.org/wiki/Morita_equivalence
same_as:
  wikidata: Q6912366
  wikipedia: Morita equivalence
---

## Summary

Two [[rings|0B1261B5ED]] are Morita equivalent when their categories of [[modules|0NZ8JA25WK]] are equivalent.
Such rings need not be isomorphic, but every property that is defined through modules alone is the same for both.
A ring $R$ and the matrix ring $M_n(R)$ are the basic example.
Morita theory describes all such equivalences through [[progenerators|ZQ9ABBKF1N]] and their endomorphism rings.

## Explanations

### L1 human

Often one studies a ring through the modules over it, much as one studies a group through its actions.
Two different rings can have module theories that look exactly alike, with the same pattern of maps between modules.
Such rings are called Morita equivalent.
For example, the field $\R$ and the ring of $2 \times 2$ real matrices have the same module theory.
A module over the matrix ring is a sum of copies of $\R^2$, and these behave just like real vector spaces.
So any question about modules can be moved from one ring to the other.