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 ![K[X]](/images/equations/SemilocalRing/Inline2.svg)
of polynomials in the indeterminate 
are the principal ideals
![<X-alpha>={f(X)(X-alpha)|f(X) in K[X]},](/images/equations/SemilocalRing/NumberedEquation1.svg) |
where 
is any element of 
.
There is a one-to-one correspondence
between these ideals and the elements of 
. Hence ![K[X]](/images/equations/SemilocalRing/Inline7.svg)
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.
