Prime ideal

In ring theory we say that an ideal $P$ of a ring $R$ is prime if $P\ne R$ and for any ideals $I,J\subseteq R$ with $IJ\subseteq P$ we have either $I\subseteq P$ or $J\subseteq P$ .

If $R$ is commutative then the following simpler definition holds: An ideal $P\subseteq R$ is prime iff $P\ne R$ and for any $a,b\in R$ if $ab\in P$ then either $a\in P$ or $b\in P$ .

This second definition easily implies the the following important property of prime ideals in commutative rings with unity:

Let $R$ be a commutative ring with unity, then an ideal $P\subseteq R$ is prime iff the quotient ring $R/P$ is an integral domain.
Prime ideals are primary and radical.

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Prime ideal