local ring

English

Noun

local ring (plural local rings)

1. (algebra) A commutative ring with a unique maximal ideal, or a noncommutative ring with a unique maximal left ideal or (equivalently) a unique maximal right ideal.

   Hyponyms: simple ring, field

   The unique maximal ideal of a (commutative) local ring contains all of the zero divisors of such ring, and all elements of the ring outside of it are units. Then in a local ring, the sum of any two zero divisors is also a zero divisor. Contrapositively, if two ring elements add up to a unit then one of them must be a unit as well. A simple example of a local ring is $\mathbb {Z} _{8}$.

2. (networking) The non-routing segment of a token ring network.

Anagrams

• carolling, collaring