idealas

Ideals are a central concept in ring theory, a branch of abstract algebra. An ideal I of a ring R is an additive subgroup of R that absorbs multiplication by elements of R: for every r in R and i in I, both ri and ir lie in I. In noncommutative rings, one distinguishes left ideals, right ideals, and two-sided ideals; a two-sided ideal is both a left and a right ideal. In a commutative ring, every ideal is automatically two-sided.

Ideals can be generated by sets. Given a subset S of R, the smallest ideal containing S is denoted (S) and is called the ideal generated by S. In a commutative ring, (S) consists of all finite sums ∑ r_j s_j with r_j in R and s_j in S. A principal ideal is generated by a single element a, written (a). In commutative rings, (a) = aR = Ra.

Examples illustrate the concept. In the ring of integers Z, every ideal is of the form nZ for some nonnegative n. In the polynomial ring k[x], the principal ideal (f) contains all multiples f g by polynomials g. In the ring Z[x], the ideal (2, x) consists of all elements 2p(x) + x q(x).

Important special classes include prime ideals and maximal ideals. A prime ideal p satisfies if ab ∈ p, then either a ∈ p or b ∈ p. A maximal ideal m is a proper ideal with no intermediate ideals between m and R; in a commutative ring with 1, every maximal ideal is prime. The radical of an ideal I, denoted sqrt(I), is the set {r ∈ R | r^n ∈ I for some n}. Quotients by ideals, R/I, inherit a ring structure and relate to kernels of surjective homomorphisms.