A maximal ideal of a ring R is an ideal I, not equal to R, such that there are no ideals "in between" I and R. In other words, if J is an ideal which contains I as a subset, then either J = I or J = R. For example, n Z is a maximal ideal of Z iff n is prime, where Z is the ring of integers. Only in a local ring is there just one maximal ideal. For instance, in the integers, a = 〈p〉 is a maximal ideal whenever p is prime. A maximal ideal m is always a prime ideal, and the quotient ring A/m is always a field. In general, not all prime ideals are maximal.

maximal ideal space | maximal ideal theorem