Redirected from Ideal
In abstract algebra, an ideal of a ring R is a subset I of R which is closed under Rlinear combinations, in a sense made precise below.

To accommodate noncommutative rings, we must distinguish three cases: left ideals, right ideals, and twosided ideals.
A subset I of the ring R is a left ideal of R if
If the ring R is commutative, then all three sorts of ideals are the same. If the ring is noncommutative, however, then they may be different.
The first two examples above are principal ideals; the principal (left) ideal generated by an element a in R is Ra := {ra : r in R}. The principal right ideal aR is defined similarly; and these two principal ideals generated by a are identical (and hence a twosided ideal) if the ring is commutative. In that case, it's common to denote the principal ideal by <a> or (a).
An ideal I is called proper if I is not equal to R. An ideal is proper if and only if it doesn't contain 1. A proper ideal is called maximal if the only proper ideal it is contained in is itself. Every ideal is contained in a maximal ideal, a consequence of Zorn's lemma. A proper ideal I is called prime if, whenever ab belongs to I, then so does a or b (or both). Every maximal ideal is prime.
Factor rings (quotient rings) and kernels
Ideals are important because they appear as the kernels of ring homomorphisms and allow one to define factor rings, as will be described next.
Recall that a function f from R to S is a ring homomorphism iff f(a + b) = f(a) + f(b) and f(ab) = f(a) f(b) for all a, b in R and f(1) = 1. Then the kernel of f is defined as
Conversely, if we start with a twosided ideal I of R, then we may define a congruence relation ~ on R as follows: a ~ b if and only if b  a is in I. In case a ~ b, we say that a and b are congruent modulo I. The equivalence class of the element a in R is given by
The map p from R to R/I defined by p(a) = a + R is a surjective ring homomorphism (or regular epimorphism[?]) whose kernel is the original ideal I. In summary, we see that ideals are precisely the kernels of ring homomorphisms.
If R is commutative and I is a maximal ideal, then the factor ring R/I is a field; if I is only a prime ideal, then R/I is only an integral domain.
The most extreme examples of factor rings are provided by modding out by the most extreme ideals, {0} and R itself. R/{0} is naturally isomorphic[?] to R, and R/R is the trivial ring[?] {0}.
The sum and the intersection of ideals is again an ideal; with these two operations as join and meet, the set of all ideals of a given ring forms a lattice.
If A is any subset of the ring R, then we can define the ideal generated by A to be the smallest ideal of R containing A; it is denoted by <A> or (A) and contains all finite sums of the form
The product of two ideals I and J is defined to be the ideal IJ generated by all products of the form ab with a in I and b in J. It is contained in the intersection of I and J.
Important properties of these ideal operations are recorded in the Noether isomorphism theorems.
Because zero belongs to it, any ideal is nonempty. In fact, property 1 in the definition can be replaced with simply the requirement that I be nonempty.
Any left, right or twosided ideal is a subgroup of the additive group (R,+).
The ring R can be considered as a left module over itself, and the left ideals of R are then seen as the submodules[?] of this module. Similarly, the right ideals are submodules of R as a right module over itself, and the twosided ideals are submodules of R as a bimodule[?] over itself. If R is commutative, then all three sorts of module are the same, just as all three sorts of ideal are the same.
The term "ideal" comes from "ideal number": ideals were seen as a generalization of the concept of number. In the ring Z of integers, every ideal can be generated by a single number (so Z is a principal ideal domain), and the ideal determines the number up to its sign. The concepts of "ideal" and "number" are therefore almost identical in Z (and in any principal ideal domain). In other rings, it turned out that the concept of "ideal" allows one to generalize several properties of numbers. For instance, in general rings one studies prime ideals instead of prime numbers, one defines coprime ideals as a generalization of coprime numbers, and one can prove a generalized Chinese remainder theorem about ideals. In a certain class of rings important in number theory, the Dedekind domains, one can even recover a version of the fundamental theorem of arithmetic: in these rings, every nonzero ideal can be uniquely written as a product of prime ideals.
Search Encyclopedia

Featured Article
