# Prime ideal

A two-sided ideal $I$ of a ring $A$ such that the inclusion $PQ\subseteq I$ for any two-sided ideals $P$ and $Q$ of $A$ implies that either $P\subseteq I$ or $Q\subseteq I$. An ideal $I$ of a ring $R$ is prime if and only if the set $R\setminus I$ is an $m$-system, i.e. for any $a,b\in R\setminus I$ there exists an $x\in R$ such that $axb\in R\setminus I$. An ideal $I$ of a ring $A$ is prime if and only if the quotient ring by it is a prime ring.

This assumes that the empty set is an $m$-system by default.