---
tag: EM7FHAFD3N
kind: concept
name: Bimodule
msc: [16D20]
sources:
  - url: https://en.wikipedia.org/wiki/Bimodule
same_as:
  wikidata: Q2903821
  wikipedia: Bimodule
---

## Summary

A bimodule over two rings $R$ and $S$ is a left [[module|0NZ8JA25WK]] over $R$ that is also a right module over $S$.
The two actions must commute.
Bimodules are the natural objects between two rings: tensoring with an $(R, S)$-bimodule turns right $R$-modules into right $S$-modules.
The two-sided ideals of a ring are exactly its sub-bimodules.
Bimodules underlie Morita theory and Hochschild cohomology.

## Explanations

### L1 human

A bimodule is acted on from both sides, by one ring on the left and by another ring on the right.
The only condition linking the two actions is that it does not matter which side acts first.
Rectangular $m \times n$ matrices are the model: square $m \times m$ matrices multiply them on the left, square $n \times n$ matrices multiply them on the right, and $(AX)B = A(XB)$.
Every ring is a bimodule over itself, with multiplication on both sides.