Ideal (ring theory)

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. 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 other integer results in another even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring similarly to the way that, in group theory, a normal subgroup can be used to construct a quotient group.What is the exact set of integers that we are forced to identify with 0? In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. 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 other integer results in another even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring similarly to the way that, 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 be distinct from 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 concept of an order ideal in order theory is derived from the notion of ideal in ring theory. A fractional ideal is a generalization of an ideal, and the usual ideals are sometimes called integral ideals for clarity. Ideals were first proposed by Richard Dedekind in 1876 in the third edition of his book Vorlesungen über Zahlentheorie (English: Lectures on Number Theory). They were a generalization of the concept of ideal numbers developed by Ernst Kummer. Later the concept was expanded by David Hilbert and especially Emmy Noether. For an arbitrary ring ( R , + , ⋅ ) {displaystyle (R,+,cdot )} , let ( R , + ) {displaystyle (R,+)} be its additive group. A subset I {displaystyle I} is called a left ideal of R {displaystyle R} if it is an additive subgroup of R {displaystyle R} that 'absorbs multiplication from the left by elements of R {displaystyle R} '; that is, I {displaystyle I} is a left ideal if it satisfies the following two conditions: A right ideal is defined with the condition 'r x ∈ I' replaced by 'x r ∈ I. A two-sided ideal is a left ideal that is also a right ideal, and is sometimes simply called an ideal. We can view a left (resp. right, two-sided) ideal of R as a left (resp. right, bi-) R-submodule of R viewed as an R-module. When R is a commutative ring, the definitions of left, right, and two-sided ideal coincide, and the term ideal is used alone. To understand the concept of an ideal, consider how ideals arise in the construction of rings of 'elements modulo'. For concreteness, let us look at the ring ℤn of integers modulo a given integer n ∈ ℤ (note that ℤ is a commutative ring). The key observation here is that we obtain ℤn by taking ℤ and wrapping it around itself so that n gets identified with 0 (since n is congruent to 0 modulo n). However, when doing so we must ensure that the resulting structure is again a ring. This requirement forces us to make some additional identifications. The notion of an ideal arises when we ask the question: The answer is the set nℤ = {nm | m∈ℤ} of all integers congruent to 0 modulo n. That is, we must wrap ℤ around itself infinitely many times so that the integers ..., n ⋅ -2, n ⋅ -1, n ⋅ +1, n ⋅ +2, ... will all align with 0. If we look at what properties this set must satisfy in order to ensure that ℤn is a ring, then we arrive at the definition of an ideal. Indeed, one can directly verify that nℤ is an ideal of ℤ. Remark. Identifications with elements other than 0 also need to be made. For example, the elements in 1 + nℤ must be identified with 1, the elements in 2 + nℤ must be identified with 2, and so on. Those, however, are uniquely determined by nℤ since ℤ is an additive group.