# Definition:Maximal Ideal of Ring

## Definition

Let $R$ be a ring.

An ideal $J$ of $R$ is maximal if and only if:

$(1): \quad J \subsetneq R$
$(2): \quad$ There is no ideal $K$ of $R$ such that $J \subsetneq K \subsetneq R$.

That is, if and only if $J$ is a maximal element of the set of all proper ideals of $R$ ordered by inclusion.

### Maximal Left Ideal

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

$(1): \quad J \subsetneq R$
$(2): \quad$ There is no left ideal $K$ of $R$ such that $J \subsetneq K \subsetneq R$.

### Maximal Right Ideal

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

$(1): \quad J \subsetneq R$
$(2): \quad$ There is no right ideal $K$ of $R$ such that $J \subsetneq K \subsetneq R$.

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

## Also defined as

Some sources insist that $R$ be a commutative ring with unity for this definition to hold.

## Also see

• Results about maximal ideals can be found here.