Editing Local ring (section)

== Definition and first consequences ==

A [[ring (mathematics)|ring]] ''R'' is a '''local ring''' if it has any one of the following equivalent properties:
* ''R'' has a unique [[maximal ideal|maximal]] left [[ring ideal|ideal]].
* ''R'' has a unique maximal right ideal.
* 1 ≠ 0 and the sum of any two non-[[unit (algebra)|unit]]s in ''R'' is a non-unit.
* 1 ≠ 0 and if ''x'' is any element of ''R'', then ''x'' or {{nowrap|1 &minus; ''x''}} is a unit.
* If a finite sum is a unit, then it has a term that is a unit (this says in particular that the empty sum cannot be a unit, so it implies 1 ≠ 0).

If these properties hold, then the unique maximal left ideal coincides with the unique maximal right ideal and with the ring's [[Jacobson radical]]. The third of the properties listed above says that the set of non-units in a local ring forms a (proper) ideal,<ref>Lam (2001), p. 295, Thm. 19.1.</ref> necessarily contained in the Jacobson radical. The fourth property can be paraphrased as follows: a ring ''R'' is local if and only if there do not exist two [[coprime]] proper ([[principal ideal|principal]]) (left) ideals, where two ideals ''I''<sub>1</sub>, ''I''<sub>2</sub> are called ''coprime'' if {{nowrap|1=''R'' = ''I''<sub>1</sub> + ''I''<sub>2</sub>}}.

<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Local Ring |url=https://mathworld.wolfram.com/LocalRing.html |access-date=2024-08-26 |website=mathworld.wolfram.com |language=en}}</ref> In the case of [[commutative ring]]s, one does not have to distinguish between left, right and two-sided ideals: a commutative ring is local if and only if it has a unique maximal ideal.
Before about 1960 many authors required that a local ring be (left and right) [[Noetherian ring|Noetherian]], and (possibly non-Noetherian) local rings were called '''quasi-local rings'''. In this article this requirement is not imposed.

A local ring that is an [[integral domain]] is called a '''local domain'''.