Definition:Maximal Ideal of Ring

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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.


Sources