> [!NOTE] Definition (Maximal Ideal) > Let $R$ be a [[Rings|ring]]. An [[Ideal of Ring|ideal]] $J$ of $R$ is maximal iff $J \neq R$ and there does not exist any proper ideal $K$ of $R$ such that $J \subsetneq K$ (that is $J$ is a [[Subsets|proper subset]] of $K$). > >That is if and only if $J$ is a maximal element of the proper ideals of $R$ ordered by the subset relation. > [!Example] > Contents