Local ring

Symbol-free definition
A commutative unital ring is termed a local ring if it satisfies the following equivalent conditions:


 * It has a unique maximal ideal
 * There is exactly one homomorphism from the commutative unital ring whose image is a field
 * There is exactly one closed point in the spectrum (corresponding to the unique maximal ideal)

Stronger properties

 * Field
 * Regular local ring
 * Completely primary ring
 * Local Artinian ring

Weaker properties

 * Semilocal ring
 * Quasilocal ring

Conjunction with other properties

 * Local Noetherian ring
 * Local Artinian ring
 * Local domain

Analogues in other algebraic structures

 * One-headed group in group theory, is a group that has a proper normal subgroup that contains every proper normal subgroup.
 * Local ring in noncommutative algebra.