maximal ideal
Noun
Meaning
maximal ideal (plural maximal ideals)
(algebra, ring theory) An ideal which cannot be made any larger (by adjoining any element to it) without making it improper (i.e., equal to the whole of the containing algebraic structure).
Related terms
Examples
If the module is Artinian, Noetherian, projective or injective, then the endomorphism ring has a unique maximal ideal, so that it is a local ring.
Moreover, these notions coincide with ring theoretic ones of prime ideal and maximal ideal in the Boolean ring A. The dual of an ideal is a filter.
More generally: the Jacobson radical of every local ring is the unique maximal ideal of the ring.
A topological space X is pseudocompact if and only if every maximal ideal in C(X) has residue field the real numbers.
Define f(x) to be the highest power of the maximal ideal M containing x (equivalently, to the power of the generator of the maximal ideal that x is associated to).
For example, the ring with as additive group and trivial multiplication (i. e. for all ) has no maximal ideal (and of course no 1): Its ideals are precisely the additive subgroups.