A commutative Noetherian unit ring having only finitely many maximal ideals. A ring having the same properties except Noetherianity is called quasilocal. If K is a field, the maximal ideals of the ring K[X] of polynomials in the indeterminate X are the principal ideals 〈X - α〉 = {f(X)(X - α)|f(X) element K[X]}, where α is any element of K. There is a one-to-one correspondence between these ideals and the elements of K. Hence K[X] is semilocal if and only if K is finite. A semilocal ring always has finite Krull dimension.

local ring | maximal ideal