(algebra) A subring which is closed under left-multiplication by any element of the ring.

(algebra) A subring which is closed under right-multiplication by any element of the ring.

(sociology) Of, or relating to an ideal type.

(algebra, ring theory) Given a commutative ring R, any ideal I such that for any a,b ∈ R, if ab ∈ I then either b ∈ I or aⁿ ∈ I for some integer n > 0.

(algebra, ring theory) A nonzero (two-sided) ideal that contains no other nonzero two-sided ideal.

(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).

(algebra, ring theory) Any (two-sided) ideal I such that for arbitrary ideals P and Q, PQ⊆I⟹P⊆I or Q⊆I.

(algebra, commutative algebra, ring theory) An ideal I within a ring R that is its own radical (i.e., for any r ∈ R, if rⁿ ∈ I for some positive integer n, then r ∈ I).

plural of ideal number

(number theory) An algebraic integer that represents an ideal in the ring of integers of a number field.