Definition:Ideal of Ring

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This page is about ideal of ring in the context of ring theory. For other uses, see ideal.

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:

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

that is, if and only if:

$\forall r \in R: J \circ r \subseteq J \land r \circ J \subseteq J$


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


Left Ideal

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

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

that is, if and only if:

$\forall r \in R: r \circ J \subseteq J$


Right Ideal

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

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

that is, if and only if:

$\forall r \in R: J \circ r \subseteq J$


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 $\struct {R, +, \circ}$ 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.

Some sources refer to such a two-sided ideal as a normal subring, in apposition with the concept of a normal subgroup.


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 of rings can be found here.


Sources

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