Prime Ideal

definitionring-theory

Definition

Given an ideal $P⊴R$ ($R$ is a commutative ring with $1_R$), we say $P$ is prime if $p\ne R$ and $ab\in P⟹a\in P$ or $b\in P$ (i.e. if $ab\in P$, $a\notin P$, then $b\in P$).

Examples

Prime Ideals of $\mathbb{Z}$

Prime ideals of $\mathbb{Z}$ are $(p)=p\mathbb{Z}$, $p_{\ge 2}$ prime integer.

Proof

If $p\in \mathbb{Z}$ is prime, $a,b\in \mathbb{Z}$, $an\in p\mathbb{Z}=(p)⟹p\mid ab\underset{\text{Euclid}}{⟹}p\mid a$ or $p\mid b⟹$ $a$ or $b$ are in $(p)⟹(p)$ prime. $(0)⊴\mathbb{Z}$ prime ideal since $0$ is the only zero-divisor in $\mathbb{Z}$. If $n\in \mathbb{Z}$, $(n)⊴\mathbb{Z}$ prime $⟹(n)\ne \mathbb{Z}⟹n\notin {\mathbb{Z}}^{\times }=\{\pm 1\}$…