Ideal (ring theory)

An ideal is an additive subgroup of a ring closed under multiplication by ring elements on the required side or sides. Ideals support quotient rings and generalize divisibility in the integers.

Ring (mathematics): A set with addition and multiplication satisfying associativity, distributivity, and additive-group laws. The ambient algebraic structure whose multiplication must preserve an ideal.
Ideal generated by a subset: The smallest ideal containing a specified subset of a ring. It formalizes how selected elements generate an ideal.
Congruence relation: An equivalence relation compatible with the operations of an algebraic structure. Congruence modulo an ideal supplies the equivalence relation behind quotient rings.
Subring: A subset of a ring that is itself a ring under the inherited operations. A subring need not absorb multiplication by every element of the ambient ring.
Additive subgroup: A subgroup of a group whose operation is written as addition. An ideal must remain closed under addition and additive inverses.
Sum of ideals: The ideal consisting of sums of elements drawn from two ideals. It combines ideals into the least ideal containing both.
Polynomial ideal: An ideal in a ring of polynomials, encoding polynomial combinations of its generators. Polynomial ideals organize systems of equations and their consequences.
Maximal ideal: A proper ideal maximal among the proper ideals of a ring. It is distinguished by maximality rather than the product test defining prime ideals.
Left ideal: An additive subgroup of a ring closed under multiplication on the left by every ring element. It is one sided form of the ideal condition in possibly noncommutative rings.
Product of ideals: The ideal formed by finite sums of products of elements from two ideals. Ideal multiplication tracks products while preserving closure.

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