---
tag: F0HQXQ0HCD
kind: concept
name: Tensor product of modules
msc: [13C05, 15A69, 16D10]
sources:
  - url: https://en.wikipedia.org/wiki/Tensor_product_of_modules
same_as:
  wikidata: Q3406761
  wikipedia: Tensor product of modules
---

## Summary

The tensor product $M \otimes_R N$ of two modules over a [[commutative ring|0310REJJY8]] $R$ is the module that turns bilinear maps on $M \times N$ into linear maps.
It is generated by the symbols $m \otimes n$, which are additive in each variable and let scalars move between the two factors.
It generalises the [[tensor product of vector spaces|0BF2DE2SYK]], but over a ring it can be much smaller than expected, even zero.
Tensor products are used for change of rings, for [[flatness|73RYYEVDKA]], and for the [[tensor-hom adjunction|AVXARAXHKY]].

## Explanations

### L1 human

The tensor product combines two modules into one, in which you can multiply an element of the first by an element of the second.
The rules are those of a product: $(m + m') \otimes n = m \otimes n + m' \otimes n$, and a scalar can be moved from one side to the other.
Sometimes everything cancels.
For example, $\Z/2\Z \otimes_\Z \Z/3\Z = 0$: since $3 \cdot 1 = 1$ in $\Z/2\Z$, one has $a \otimes b = 3a \otimes b = a \otimes 3b = a \otimes 0 = 0$.