Ideal

From AoPSWiki

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 $A$ is a ring, a subset $\mathfrak{a}$ of $A$ is called a left ideal of $A$ if it is a subgroup under addition, and if $xa \in \alpha$ , for all $x\in R$ and $a\in \mathfrak{a}$ . Symbolically, this can be written as $0\in \mathfrak{a}, \qquad \mathfrak{a+a\subseteq a}, \qquad A \mathfrak{a \subseteq a} .$ A right ideal is defined similarly, but with the modification $\mathfrak{a}A \subseteq \mathfrak{a}$ . If $\mathfrak{a}$ 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 $A$ are exactly the left ideals of the opposite ring $A^0$ .

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 $\mathbb{Z}$ , the ideals are the rings of the form $n \mathbb{Z}$ , for some integer $n$ .

In a field $F$ , the only ideals are the set $\{0\}$ and $F$ itself.

In general, if $A$ is a ring and $x$ is an element of $A$ , the set $Ax$ is a left ideal of $A$ . Ideals of this form are known as principal ideals.

By abuse of language, a (left, right, two-sided) ideal of a ring $A$ is called maximal if it is a maximal element of the set of (left, right, two-sided) ideals distinct from $A$ . A two-sided ideal $\mathfrak{m}$ is maximal if and only if its quotient ring $A/\mathfrak{m}$ is a field.

An ideal $\mathfrak{p}$ is called a prime ideal if $ab\in \mathfrak{p}$ implies $a\in \mathfrak{p}$ or $b \in \mathfrak{p}$ . A two-sided ideal $\mathfrak{p}$ is prime if and only if its quotient ring $A/ \mathfrak{p}$ is a domain. In commutative algebra, the notion of prime ideal is central; it generalizes the notion of prime numbers in $\mathbb{Z}$ .

Generated Ideals

Let $A$ be a ring, and let $(x_i)_{i\in I}$ be a family of elements of $A$ . The left ideal generated by the family $(x_i)_{i\in I}$ is the set of elements of $A$ of the form $\sum_{i \in I} a_i x_i,$ where $(a_i)_{i \in I}$ is a family of elements of $A$ of finite support, as this set is a left ideal of $A$ , thanks to distributivity, and every element of the set must be in every left ideal containing $(x_i)_{i\in I}$ . Similarly, the two-sided ideal generated by $(x_i)_{i\in I}$ is the set of elements of $A$ of the form $\sum_{i\in I} a_i x_i b_i,$ where $(a_i)_{i\in I}$ and $(b_i)_{i \in I}$ are families of finite support.

The two-sided ideal generated by a finite family ${a_1, \dotsc, a_n}$ is often denoted $(a_1, \dotsc, a_n)$ .

If $(\mathfrak{a}_i)_{i\in I}$ is a set of (left, right, two-sided) ideals of $A$ , then the (left, two sided) ideal generated by $\bigcup_{i\in I} \mathfrak{a}_i$ is the set of elements of the form $\sum_i x_i$ , where $x_i$ is an element of $\mathfrak{a}_i$ and $(x_i)_{i\in I}$ is a family of finite support. For this reason, the ideal generated by the $\mathfrak{a}_i$ is sometimes denoted $\sum_{i\in I} \mathfrak{a}_i$ .