A commutative Noetherian unit ring having only finitely many maximal ideals. A ring having the same properties except Noetherianity is called quasilocal.
If is a field, the maximal ideals of the ring of polynomials in the indeterminate are the principal ideals
where is any element of . There is a one-to-one correspondence between these ideals and the elements of . Hence is semilocal if and only if is finite.
A semilocal ring always has finite Krull dimension.
The ring of integers is an example of a Noetherian nonsemilocal ring, since its maximal ideals are the principal ideals , where is any prime number.