Ideal

An ideal is an additive subgroup of a ring closed under multiplication by every ring element. Ideals encode which elements can be treated as multiples of a chosen subset.

Ring (mathematics): A set with addition and multiplication satisfying associativity, distributivity, and additive-group laws. Ideals are defined inside rings and inherit their addition.
Ideal generated by a set: The smallest ideal containing a specified subset of a ring. It constructs ideals from chosen ring elements by closing under required operations.
Polynomial ring: A ring whose elements are polynomials with coefficients in a given ring. Polynomial ideals encode equations and relations among variables.
Left ideal: An additive subgroup closed under multiplication on the left by every ring element. In noncommutative rings, left closure alone need not imply right closure.
Additive subgroup: A subset of a group that is itself a group under the same operation. An ideal must be an additive subgroup of its ring.
Sum of ideals: The ideal consisting of sums of elements drawn from two ideals. Adding ideals combines their constraints into the smallest ideal containing both.
Algebraic variety: A geometric set defined as the common zeros of polynomial equations. Polynomial ideals collect equations defining algebraic varieties.
Right ideal: An additive subgroup closed under multiplication on the right by every ring element. It separates one-sided closure from the two-sided condition in the definition.
Principal ideal: An ideal generated by a single element of a ring. It is the simplest form of ideal when one element generates all members.
Product of ideals: The ideal formed by finite sums of products of elements from two ideals. Multiplication combines ideal conditions while preserving ideal structure.

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