Definition:Ideal of Ring

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Definition

Let $\left({R, +, \circ}\right)$ be a ring.

Let $\left({J, +}\right)$ be a subgroup of $\left({R, +}\right)$.


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.


Also see

  • Results about ideals can be found here.


Sources