Ideals (ring theory)

additive subgroup of a ring closed under multiplcation by arbitrary ring element

proper ideal such that the only ideal it is properly contained in is the ring itself

ideal generated by one element

concept in algebra

proper ideal q such that, whenever xy ∈ q, then either x ∈ q, or some power of y is in q

radical of a ring as a module over itself; the intersection of all maximal right (or equivalently left) ideals of the ring; the sum of all superfluous right (or equiv. left) ideals

mathematical set in algebraic number theory

generalization of the ring-theoretical notion of ideal to integral domains

set of the nilpotent elements of a ring

mathematical theorem of dimensional theory

theorem stating that a nonzero ring has a maximal left (or right) ideal

in order theory, a nonempty, upward‐directed, downward‐closed subset of a preordered set

concept in module theory

generalizations of prime ideals and prime rings

ideal such that a sufficiently high power of it is zero

ideal consisting solely of nilpotent elements

annihilator of a simple module

the ideal-theoretic generalization of the field norm

the theorem that extending ideals gives a mapping on the class group of an algebraic number field to the class group of its Hilbert class field, which sends all ideal classes to the class of a principal ideal