a characterization of the radical of an ideal

Proof.

Suppose $x\in\sqrt{I}$, and $P$ is a prime ideal containing $I$. Then $R-P$ is an $m$-system (http://planetmath.org/MSystem). If $x\in R-P$, then $(R-P)\cap I\neq\varnothing$, contradicting the assumption  that $I\subseteq P$. Therefore $x\notin R-P$. In other words, $x\in P$, and we have one of the inclusions.

Conversely, suppose $x\notin\sqrt{I}$. Then there is an $m$-system $S$ containing $x$ such that $S\cap I=\varnothing$. Enlarge $I$ to a prime ideal $P$ disjoint from $S$, so that $x\notin P$ (we can do this; for a proof, see the second remark in this entry (http://planetmath.org/MSystem)). By contrapositivity, we have the other inclusion. ∎

Remark. This shows that every prime ideal is a radical ideal: for $\sqrt{P}$ is the intersection of all prime ideals containing $P$, and if $P$ is itself prime, then $P=\sqrt{P}$.

Title a characterization of the radical of an ideal ACharacterizationOfTheRadicalOfAnIdeal 2013-03-22 18:04:50 2013-03-22 18:04:50 CWoo (3771) CWoo (3771) 7 CWoo (3771) Derivation msc 16N40 msc 13-00 msc 14A05