Maximal Ideal

A maximal ideal of a Ring $R$ is an Ideal $I$, not equal to $R$, such that there are no Ideals ``in between'' $I$ and $R$. In other words, if $J$ is an Ideal which contains $I$ as a Subset, then either $J=I$ or $J=R$. For example, $n\Bbb{Z}$ is a maximal ideal of $\Bbb{Z}$ Iff $n$ is Prime, where $\Bbb{Z}$ is the Ring of Integers.

See also Ideal, Prime Ideal, Regular Local Ring, Ring