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

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

(algebra, ring theory) Given an integral domain R and its field of fractions K = Frac(R), an R-submodule I of K such that for some nonzero r∈R, rI ⊆ R.

(algebra, ring theory) For given ring R, any ideal I such that Q = R / I, the set of cosets of elements of I in R, is a ring (the quotient ring of I in R).

(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) 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, 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).

Alternative form of ideaed

Alternative form of beau idéal