You are here
Home$m$system
Primary tabs
$m$system
Let $R$ be a ring. A subset $S$ of $R$ is called an $m$system if

$S\neq\varnothing$, and

for every two elements $x,\,y\in S$, there is an element $r\in R$ such that $xry\in S$.
$m$Systems are a generalization of multiplicatively closet subsets in a ring. Indeed, every multiplicatively closed subset of $R$ is an $m$system: any $x,y\in S$, then $xy\in S$, hence $xyy\in S$. However, the converse is not true. For example, the set
$\{r^{n}\mid r\in R\mbox{ and }n\mbox{ is an odd positive integer}\}$ 
is an $m$system, but not multiplicatively closed in general (unless, for example, if $r=1$).
Remarks. $m$Systems and prime ideals of a ring are intimately related. Two basic relationships between the two notions are
1. 2. Given an $m$system $S$ of $R$ and an ideal $I$ with $I\cap S=\varnothing$. Then there exists a prime ideal $P\subseteq R$ with the property that $P$ contains $I$ and $P\cap S=\varnothing$, and $P$ is the largest among all ideals with this property.
Proof.
Let $\mathcal{C}$ be the collection of all ideals containing $I$ and disjoint from $S$. First, $I\in\mathcal{C}$. Second, any chain $K$ of ideals in $\mathcal{C}$, its union $\bigcup K$ is also in $\mathcal{C}$. So Zorn’s lemma applies. Let $P$ be a maximal element in $\mathcal{C}$. We want to show that $P$ is prime. Suppose otherwise. In other words, $aRb\subseteq P$ with $a,b\notin P$. Then $\langle P,a\rangle$ and $\langle P,b\rangle$ both have nonempty intersections with $S$. Let
$c=p+fag\in\langle P,a\rangle\cap S\quad\mbox{ and }\quad d=q+hbk\in\langle P,b% \rangle\cap S,$ where $p,q\in P$ and $f,g,h,k\in R$. Then there is $r\in R$ such that $crd\in S$. But this implies that
$crd=(p+fag)r(q+hbk)=p(rq+rhbk)+(fagr)q+f\big(a(grh)b\big)k\in P$ as well, contradicting $P\cap S=\varnothing$. Therefore, $P$ is prime. ∎
$m$Systems are also used to define the noncommutative version of the radical of an ideal of a ring.
Mathematics Subject Classification
16U20 no label found13B30 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
 Corrections
Comments
m systems and prime ideals
unless the empty set is considered an m system the ring R which is a prime ideal is an exception in item 1.