In ring theory, an ideal is a special kind of subset of a ring. Two-sided ideals in rings are the kernels of ring homomorphisms; in this way, two-sided ideals of rings are similar to normal subgroups of groups.
Specifially, if is a ring, a subset of is called a left ideal of if it is a subgroup under addition, and if , for all and . Symbolically, this can be written as A right ideal is defined similarly, but with the modification . If is both a left ideal and a right ideal, it is called a two-sided ideal. In a commutative ring, all three kinds of ideals are the same; they are simply called ideals. Note that the right ideals of a ring are exactly the left ideals of the opposite ring .
An ideal has the structure of a pseudo-ring, that is, a structure that satisfies the properties of rings, except possibly for the existence of a multiplicative identity.
Examples of Ideals; Types of Ideals
In the ring , the ideals are the rings of the form , for some integer .
In a field , the only ideals are the set and itself.
In general, if is a ring and is an element of , the set is a left ideal of . Ideals of this form are known as principal ideals.
By abuse of language, a (left, right, two-sided) ideal of a ring is called maximal if it is a maximal element of the set of (left, right, two-sided) ideals distinct from . A two-sided ideal is maximal if and only if its quotient ring is a field.
An ideal is called a prime ideal if implies or . A two-sided ideal is prime if and only if its quotient ring is a domain. In commutative algebra, the notion of prime ideal is central; it generalizes the notion of prime numbers in .
Let be a ring, and let be a family of elements of . The left ideal generated by the family is the set of elements of of the form where is a family of elements of of finite support, as this set is a left ideal of , thanks to distributivity, and every element of the set must be in every left ideal containing . Similarly, the two-sided ideal generated by is the set of elements of of the form where and are families of finite support.
The two-sided ideal generated by a finite family is often denoted .
If is a set of (left, right, two-sided) ideals of , then the (left, two sided) ideal generated by is the set of elements of the form , where is an element of and is a family of finite support. For this reason, the ideal generated by the is sometimes denoted .
Multiplication of Ideals
If and are two-sided ideals of a ring , then the set of elements of the form , for and , is also an ideal of . It is called the product of and , and it is denoted . It is generated by the elements of the form , for and . Since and are two-sided ideals, is a subset of both and of , so
Proposition 1. Let and be two-sided ideals of a ring such that , for each index . Then
Proof. We induct on . For , the proposition is degenerately true.
Now, suppose the proposition holds for . Then which proves the proposition.
Proposition 2. Let be two-sided ideals of a ring such that , for any distinct indices and . Then where is the symmetric group on .
Proof. It is evident that We prove the converse by induction on .
For , the statement is trivial. For , we note that 1 can be expressed as , where . Thus for any ,
Now, suppose that the statement holds for the integer . Then by the previous proposition, so from the case , as desired.
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