Ideales

Ideales is the plural form used in several languages, such as Spanish and Portuguese, for the mathematical concept of an ideal in ring theory. An ideal I of a ring R is a subset that is an additive subgroup of R and is closed under multiplication by elements of R: for all a, b in I and r in R, a + b is in I and ar, ra are in I. If R is commutative, left and right ideals coincide, and one speaks simply of ideals. In noncommutative rings, one distinguishes left, right, and two-sided ideals.

Principal ideals generated by an element a in R are denoted (a) = {ra : r in R}; in a commutative ring, (a) = aR. The collection of all ideals of R forms a partially ordered set under inclusion and has a lattice structure, reflecting how ideals intersect and sum.

Quotients: for an ideal I, the quotient ring R/I consists of cosets and inherits addition and multiplication. The projection map π: R → R/I is a surjective ring homomorphism with kernel I. The fundamental homomorphism theorem states that R/ker φ ≅ im φ for any ring homomorphism φ.

Prime and maximal ideals are central to the structure theory of rings. A proper ideal P is prime if whenever ab ∈ P, then a ∈ P or b ∈ P. An ideal M is maximal if there is no strictly larger proper ideal containing M. In a commutative ring with unity, every maximal ideal is prime, and prime ideals correspond to irreducible algebraic objects in algebraic geometry via the spectrum of R.

Beyond pure algebra, ideals are used in number theory (factorization of ideals in Dedekind domains) and algebraic geometry, as well as in computational algebra for solving polynomial systems.