Semilocal Ring

A semilocal ring is a commutative ring with multiplicative identity having only finitely many maximal ideals (Atiyah and Macdonald 1969, p. 4). No Noetherian hypothesis is required.

For example, the ring

where denotes the greatest common divisor, has exactly two maximal ideals, and . It is therefore semilocal but not a local ring. This follows from the correspondence of prime ideals under localization (Atiyah and Macdonald 1969, p. 41).

In contrast, for every field , the polynomial ring is not semilocal. Its maximal ideals are generated by irreducible polynomials, which need not be linear, and there are infinitely many such maximal ideals even when is a finite field.

A Noetherian semilocal ring has finite Krull dimension, since its finitely many maximal ideals each have finite height (Atiyah and Macdonald 1969, p. 121).

The ring of integers is another example of a Noetherian nonsemilocal ring, since its maximal ideals are the principal ideals , where is any prime number.