---
tag: CQ3616BH5C
kind: definition
name: Morita equivalence
msc: [16D90]
defines: RD5WKA1VK7
uses: [0B1261B5ED, 0NZ8JA25WK, K7A04ZBD92]
sources:
  - url: https://en.wikipedia.org/wiki/Morita_equivalence
---

## Definition

Let $R$ and $S$ be [[rings|0B1261B5ED]] with identity.
Let $R\text{-Mod}$ be the category of left $R$-[[modules|0NZ8JA25WK]] and $R$-linear maps, and likewise $S\text{-Mod}$.
The rings $R$ and $S$ are *Morita equivalent* if there is an [[equivalence of categories|K7A04ZBD92]] $F \colon R\text{-Mod} \to S\text{-Mod}$.

## Remarks

Example: for a field $k$, the rings $k$ and $M_2(k)$ are Morita equivalent.
The functor sends a vector space $V$ to $V^2 = k^2 \otimes_k V$, the column vectors with entries in $V$, on which $M_2(k)$ acts by matrix multiplication.
A quasi-inverse sends a module $N$ to $e_{11} N$, where $e_{11}$ is the matrix unit.
Non-example: $k$ and $k \times k$ are not Morita equivalent.
In $k\text{-Mod}$ every nonzero module has a nonzero map to every other nonzero module, while the simple $k \times k$-modules $k \times 0$ and $0 \times k$ have no nonzero map between them.
An equivalence of categories preserves this property.
An equivalence between module categories is automatically additive and is naturally isomorphic to $P \otimes_R -$ for an $S$-$R$-bimodule $P$.
Morita's theorem: $R$ and $S$ are Morita equivalent if and only if $S \cong \End_R(P)^{\mathrm{op}}$ for some [[progenerator|ZQ9ABBKF1N]] $P$ of $R\text{-Mod}$, where $\End_R(P)$ acts on $P$ from the left.
The definition with right modules gives the same relation.
Isomorphic rings are Morita equivalent, and for commutative rings the converse holds.