---
tag: SZHY975FFX
kind: concept
name: Reduced ring
msc: [13A99, 13A15]
specializes: [0310REJJY8]
sources:
  - url: https://en.wikipedia.org/wiki/Reduced_ring
same_as:
  wikidata: Q6423024
  wikipedia: Reduced ring
---

## Summary

A reduced ring is a ring with no nonzero [[nilpotent elements|0N9R5NN6TW]].
For a commutative ring this means that its [[nilradical|QQDN6EYQNJ]] is zero.
Every [[integral domain|09S3PFBEPM]] is reduced, and the coordinate rings of algebraic varieties are reduced.
Dividing a commutative ring by its nilradical always gives a reduced ring.

## Explanations

### L1 human

In a reduced ring, if some power of an element is zero, then the element is already zero.
The ring $\Z/6\Z$ is reduced, while $\Z/4\Z$ is not, because $2 \ne 0$ but $2^2 = 4 = 0$ there.
A reduced ring may still have zero divisors: in $\Z/6\Z$, $2 \cdot 3 = 0$.
Geometrically, a reduced ring is a ring of functions in which a function that vanishes everywhere is zero.