Maximal ideal

Definition

Let $R$ be a ring. An ideal $M$ of $R$ is maximal if it has these two properties.

1. $M \neq R$.
2. For every ideal $I$ of $R$ with $M \subseteq I$, either $I = M$ or $I = R$.

Remarks

The definition is the same for left, right and two-sided ideals, with ideals of the same type in condition 2. In $\mathbb{Z}$, the ideal $n\mathbb{Z}$ with $n \ge 0$ is maximal if and only if $n$ is a prime number. In a commutative ring $R$, an ideal $M$ is maximal if and only if $R/M$ is a field. The zero ideal of a field is its only maximal ideal.

Explanations

L1 human

A maximal ideal is as large as possible without being the whole ring. In the integers, $(6)$ is not maximal, because $(6) \subseteq (2)$. The ideal $(2)$ is maximal, because no ideal lies strictly between $(2)$ and $\mathbb{Z}$.