In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or prime.

A ring $R$ is a local ring if it has any one of the following equivalent properties:

(1) $R$ has a unique maximal left ideal. (2) $R$ has a unique maximal right ideal. (3) $1 \neq 0$ and the sum of any two non-units in $R$ is a non-unit. (4) $1\neq 0$ and if $x$ is any element of $R$, then $x$ or $1- x$ is a unit. (5) 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\neq 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.