---
tag: GRWE9JJ5CT
kind: concept
name: Morita invariant
msc: [16D90]
sources:
  - url: https://en.wikipedia.org/wiki/Morita_equivalence
---

## Summary

A Morita invariant is a property of [[rings|0B1261B5ED]], or an object attached to rings, that is the same for any two [[Morita equivalent|RD5WKA1VK7]] rings.
Properties defined through the category of modules are Morita invariant, such as being semisimple, left Artinian or left Noetherian.
Properties defined through elements, such as commutativity or having no zero divisors, are usually not.
Morita invariants let one replace a ring by a simpler equivalent one, such as $M_n(R)$ by $R$.

## Explanations

### L1 human

Two rings are Morita equivalent when their modules behave the same way.
A Morita invariant is a feature that cannot tell such rings apart.
For example, $\R$ and the ring of $2 \times 2$ real matrices are Morita equivalent, and both are semisimple, so semisimplicity is a candidate invariant, and indeed it is one.
But $\R$ is commutative and the matrix ring is not, so commutativity is not a Morita invariant.