Local Ring - Definition and First Consequences

Definition and First Consequences

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

• R has a unique maximal left ideal.
• R has a unique maximal right ideal.
• 1 ≠ 0 and the sum of any two non-units in R is a non-unit.
• 1 ≠ 0 and if x is any element of R, then x or 1 − x is a unit.
• If a finite sum is a unit, then so are some of its terms (in particular the empty sum is not a unit, hence 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, 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) (left) ideals where two ideals I₁, I₂ are called coprime if R = I₁ + I₂.

In the case of commutative rings, 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.

Some authors require that a local ring be (left and right) Noetherian, and the non-Noetherian rings are then 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.