# Definition:Ideal of Ring

## Contents

## Definition

Let $\struct {R, +, \circ}$ be a ring.

Let $\struct {J, +}$ be a subgroup of $\struct {R, +}$.

Then $J$ is an **ideal of $R$** if and only if:

- $\forall j \in J: \forall r \in R: j \circ r \in J \land r \circ j \in J$

The letter $J$ is frequently used to denote an **ideal**.

### Left Ideal

$J$ is a **left ideal of $R$** if and only if:

- $\forall j \in J: \forall r \in R: r \circ j \in J$

### Right Ideal

$J$ is a **right ideal of $R$** if and only if:

- $\forall j \in J: \forall r \in R: j \circ r \in J$

It follows that in a commutative ring, a left ideal, a right ideal and an **ideal** are the same thing.

### Proper Ideal

A **proper ideal** $J$ of $\left({R, +, \circ}\right)$ is an ideal of $R$ such that $J$ is a proper subset of $R$.

That is, such that $J \subseteq R$ and $J \ne R$.

## Also known as

An **ideal** can also be referred to as a **two-sided ideal** to distinguish it from a **left ideal** and a **right ideal**.

Some sources use $I$ to denote an **ideal**, but on $\mathsf{Pr} \infty \mathsf{fWiki}$ this can be too easily conflated with an identity mapping.

## Examples

### Set of Even Integers

The set $2 \Z$ of even integers forms an ideal of the ring of integers.

### Order 2 Matrices with some Zero Entries

Let $R$ be the set of all order $2$ square matrices of the form $\begin{pmatrix} x & y \\ 0 & z \end{pmatrix}$ with $x, y, z \in \R$.

Let $S$ be the set of all order $2$ square matrices of the form $\begin{pmatrix} x & y \\ 0 & 0 \end{pmatrix}$ with $x, y \in \R$.

Then $R$ is a ring and $S$ is an ideal of $R$.

## Also see

- Results about
**ideals**can be found here.

## Sources

- 1955: John L. Kelley:
*General Topology*... (previous) ... (next): Chapter $0$: Algebraic Concepts - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 22$ - 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $5$: Rings: $\S 21$. Ideals - 1970: B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*... (previous) ... (next): $\S 2.2$: Homomorphisms: Definition $2.5$ - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 6$: Rings and fields - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 58$. Ideals