---
tag: G3NN0YEVVG
kind: definition
name: Morita invariant
msc: [16D90]
defines: GRWE9JJ5CT
uses: [0B1261B5ED, RD5WKA1VK7]
sources:
  - url: https://en.wikipedia.org/wiki/Morita_equivalence
---

## Definition

Let $P$ be a property of [[rings|0B1261B5ED]] with identity.
The property $P$ is a *Morita invariant* if the following holds.
Whenever $R$ and $S$ are [[Morita equivalent|RD5WKA1VK7]] rings, $R$ has $P$ if and only if $S$ has $P$.
More generally, an assignment of an object $X(R)$ to each ring $R$ is a Morita invariant if $X(R) \cong X(S)$ whenever $R$ and $S$ are Morita equivalent.

## Remarks

Example: being left Noetherian is a Morita invariant.
A ring is left Noetherian if and only if every finitely generated left module satisfies the ascending chain condition on submodules.
Finite generation, submodules and chains of submodules are preserved by any equivalence of module categories.
Other Morita invariant properties are being simple, semisimple, left Artinian, primitive, prime, left hereditary, and having a given left global dimension.
The [[center|MVT64B3B6H]] of a ring, the lattice of its two-sided ideals, and its Grothendieck group $K_0$ are Morita invariant objects.
Non-example: commutativity is not a Morita invariant, since a commutative ring $R \ne 0$ is Morita equivalent to $M_2(R)$, which is not commutative.
For the same reason, being a domain and being a local ring are not Morita invariants.