---
tag: K5Z67HR9ET
kind: concept
name: Total ring of fractions
msc: [13B30]
specializes: [4C1AE84N5N]
sources:
  - url: https://en.wikipedia.org/wiki/Total_ring_of_fractions
same_as:
  wikidata: Q5695923
  wikipedia: Total ring of fractions
---

## Summary

The total ring of fractions of a [[commutative ring|0310REJJY8]] $R$ is the [[localization|4C1AE84N5N]] of $R$ at the set of elements that are not [[zero divisors|0NNRDW10RZ]].
It is the largest localization in which $R$ still embeds.
For an [[integral domain|0QVD7WBFGN]], it is the [[field of fractions|0F2PTGX1A2]].

## Explanations

### L1 human

To build fractions from a ring, you want to divide by as many elements as possible.
You cannot divide by a zero divisor $s$, with $sa = 0$ for some $a \ne 0$, without making $a$ equal to zero.
So the safe choice is to allow every element that is not a zero divisor as a denominator.
For the integers this gives the rational numbers.
The [[definition|8MGH8PGJWR]] states it.