idéals

Ideals are a central concept in ring theory, a branch of abstract algebra. Given a ring R, an ideal I is a subset that is closed under addition and absorbs multiplication by any element of R: for all a, b in I and r in R, a + b belongs to I and ra belongs to I (and in commutative rings, ar also belongs to I). If R is not commutative, one distinguishes left ideals, right ideals and two-sided ideals; a two-sided ideal is closed under multiplication by any element of R on both sides.

In examples, in the ring of integers Z every ideal is principal and has the form nZ for some nonnegative integer n. In a polynomial ring k[x] over a field k, the ideal generated by a polynomial f is denoted (f) and consists of all multiples f·g. More generally, ideals can be generated by sets of elements, with the smallest ideal containing a given subset S of R.

Quotients: If I is an ideal of R, the quotient ring R/I encodes congruence relations modulo I. The natural projection map R → R/I sends r to its coset r + I. The quotient helps study the structure of R via simpler rings.

Prime and maximal ideals: An ideal P is prime if R/P is an integral domain, and maximal if R/M is a field. Every maximal ideal is prime in a commutative ring with unity. The intersection of prime ideals is the nilradical, relating to nilpotent elements.

Operations and structure: Ideals form a lattice under inclusion, with sum I + J and product IJ defined by finite sums of elements from I and J. In Noetherian rings, ideals satisfy the ascending chain condition; Hilbert’s basis theorem ensures polynomial rings over Noetherian rings are Noetherian. Concepts such as primary decomposition and radical ideals connect ideals to geometry and algebraic structures.

Applications span algebraic geometry, module theory, and number theory, where ideals model constraints, factorization, and quotient structures.