# Prime ideal

From Encyclopedia of Mathematics

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.

#### Comments

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

#### References

[a1] | L.H. Rowen, "Ring theory" , I , Acad. Press (1988) pp. 163 |

**How to Cite This Entry:**

Prime ideal.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Prime_ideal&oldid=39153

This article was adapted from an original article by K.A. Zhevlakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article