Ideal (ring theory)

In mathematics, and more specifically in ring theory, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer (even or odd) results in an even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring in a way similar to how, in group theory, a normal subgroup can be used to construct a quotient group.

Among the integers, the ideals correspond one-for-one with the non-negative integers: in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number. However, in other rings, the ideals may not correspond directly to the ring elements, and certain properties of integers, when generalized to rings, attach more naturally to the ideals than to the elements of the ring. For instance, the prime ideals of a ring are analogous to prime numbers, and the Chinese remainder theorem can be generalized to ideals. There is a version of unique prime factorization for the ideals of a Dedekind domain (a type of ring important in number theory).

The related, but distinct, concept of an ideal in order theory is derived from the notion of an ideal in ring theory. A fractional ideal is a generalization of an ideal, and the usual ideals are sometimes called integral ideals for clarity.

Ernst Kummer invented the concept of ideal numbers to serve as the "missing" factors in number rings in which unique factorization fails; here the word "ideal" is in the sense of existing in imagination only, in analogy with "ideal" objects in geometry such as points at infinity. In 1876, Richard Dedekind replaced Kummer's undefined concept by concrete sets of numbers, sets that he called ideals, in the third edition of Dirichlet's book Vorlesungen über Zahlentheorie, to which Dedekind had added many supplements. Later the notion was extended beyond number rings to the setting of polynomial rings and other commutative rings by David Hilbert and especially Emmy Noether.

Given a ring ${\displaystyle R}$, a left ideal is a subset ${\displaystyle I}$ of ${\displaystyle R}$ that is a subgroup of the additive group of ${\displaystyle R}$ that is closed under left multiplication by elements of ⁠${\displaystyle R}$⁠; that is, for every ${\displaystyle r\in R}$ and every ⁠${\displaystyle x,y\in I}$⁠, one has:

In other words, a left ideal is a left submodule of ${\displaystyle R}$, considered as a left module over itself.

A right ideal is defined similarly, with the condition ${\displaystyle rx\in I}$ replaced by ⁠${\displaystyle xr\in I}$⁠. A two-sided ideal is a left ideal that is also a right ideal.

If the ring is commutative, the definitions of left, right, and two-sided ideal coincide, and one talks simply of an ideal. In the non-commutative case, "ideal" is often used instead of "two-sided ideal".