Ideals

Introduction

In group theory, normal subgroups are the special subgroups that allow for the construction of quotient groups. In ring theory, the analogous concept is that of an ideal. Ideals play a fundamental role in ring theory and provide the foundation for constructing quotient rings.

Definition

Definition 9.1: A subset $I$ of a ring $R$ is a two-sided ideal if it satisfies:

$(I, +)$ is a subgroup of $(R, +)$
For every $r \in R$ and every $x \in I$ , both $r x$ and $x r$ are in $I$

This second condition is called the absorption property. For commutative rings, left and right ideals are the same.

Properties

Kernel of Homomorphisms

Just as normal subgroups are the kernels of group homomorphisms, two-sided ideals are precisely the kernels of ring homomorphisms. This property allows for the construction of quotient rings.

Generated Ideals

If $S$ is a subset of a ring $R$ , the ideal generated by $S$ , denoted $(S)$ , is the smallest ideal containing $S$ . It consists of all finite linear combinations of elements of $S$ with coefficients in $R$ .

Principal Ideals

An ideal $I$ is principal if it is generated by a single element, i.e., $I = (a)$ for some $a \in R$ .

Examples

Example 1: Ideals in $Z$

The ideals in $Z$ are precisely the sets $n Z = {n k ∣ k \in Z}$ for some $n \in Z$ . These are all principal ideals.

Example 2: Ideals in Polynomial Rings

In $R [x]$ , the ideal $(x^{2} + 1)$ consists of all polynomials that are multiples of $x^{2} + 1$ .

Example 3: Zero Ideal and Unit Ideal

The zero ideal $(0) = {0}$ is always an ideal
The unit ideal $(1) = R$ is the entire ring

Example 4: Maximal Ideals

A maximal ideal is a proper ideal that is not contained in any other proper ideal. In $Z$ , the maximal ideals are $(p)$ where $p$ is prime.

Types of Ideals

Prime Ideals

An ideal $P$ is prime if for any $a, b \in R$ , whenever $a b \in P$ , then either $a \in P$ or $b \in P$ .

Examples:

In $Z$ , the prime ideals are $(0)$ and $(p)$ where $p$ is prime
In $R [x]$ , the ideal $(x^{2} + 1)$ is prime

Maximal Ideals

An ideal $M$ is maximal if there is no other ideal $I$ such that $M \subset I \subset R$ .

Examples:

In $Z$ , the maximal ideals are $(p)$ where $p$ is prime
In $R [x]$ , the ideal $(x - a)$ is maximal for any $a \in R$

Radical Ideals

The radical of an ideal $I$ , denoted $\sqrt{I}$ , is the set of all elements $r \in R$ such that $r^{n} \in I$ for some positive integer $n$ .

An ideal $I$ is radical if $I = \sqrt{I}$ .

Operations on Ideals

Sum of Ideals

The sum of two ideals $I$ and $J$ is the ideal $I + J = {i + j ∣ i \in I, j \in J}$ .

Product of Ideals

The product of two ideals $I$ and $J$ is the ideal $I J$ generated by all products $i j$ where $i \in I$ and $j \in J$ .

Intersection of Ideals

The intersection of two ideals $I$ and $J$ is the ideal $I \cap J$ .

Applications

Application 1: Quotient Rings

Ideals are essential for constructing quotient rings, which are fundamental in algebra.

Application 2: Algebraic Geometry

Ideals correspond to geometric objects, establishing a deep connection between algebra and geometry.

Application 3: Number Theory

Ideals in rings of integers are crucial for algebraic number theory.