---
tag: MVT64B3B6H
kind: concept
name: Center of a ring
msc: [16U70]
sources:
  - url: https://en.wikipedia.org/wiki/Center_(ring_theory)
same_as:
  wikidata: Q30603786
  wikipedia: Center (ring theory)
---

## Summary

The center of a [[ring|0B1261B5ED]] is the set of elements that commute with every element of the ring.
It is a commutative subring, and the ring is commutative exactly when its center is the whole ring.
Every ring is an algebra over its center, which is why the center is the natural base ring in the study of noncommutative rings.
It is the ring analogue of the [[centre of a group|0DZ93493VN]].

## Explanations

### L1 human

Some elements of a ring can be moved past any other element in a product without changing it.
These elements form the center.
In the ring of $n \times n$ real matrices, only the multiples of the identity matrix commute with every matrix, so the center is a copy of $\R$.
In the [[quaternions|0GE4DVQKR8]], the center consists of the real numbers.