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$m$-system (Definition)

Let $ R$ be a ring. A subset $ S$ of $ R$ is called an $ m$-system if

  • $ S\ne \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

$\displaystyle \lbrace r^n\mid r\in R$    and $\displaystyle n$    is an odd positive integer$\displaystyle \rbrace$
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. An ideal $ P$ in a ring $ R$ is a prime ideal iff $ R-P$ is an $ m$-system.
    Proof. $ P$ is prime iff $ xRy\subseteq P$ implies $ x$ or $ y\in P$, iff $ x,y\in R-P$ implies that there is $ r\in R$ with $ xry\notin P$ iff $ R-P$ is an $ m$-system. $ \qedsymbol$
  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 non-empty intersections with $ S$. Let
    $\displaystyle c=p+fag \in \langle P,a\rangle \cap S$    and $\displaystyle \quad d=q+hbk \in \langle P,b\rangle \cap S,$
    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
    $\displaystyle crd = (p+fag)r(q+hbk)= p(rq+rhbk)+(fagr)q+f\big(a(grh)b\big)k \in P$
    as well, contradicting $ P\cap S=\varnothing$. Therefore, $ P$ is prime. $ \qedsymbol$
$ m$-Systems are also used to define the non-commutative version of the radical of an ideal of a ring.



"$m$-system" is owned by CWoo. [ full author list (2) ]
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See Also: multiplicatively closed, $n$-system, prime ideal

Other names:  m-system
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Cross-references: radical of an ideal, non-commutative, intersections, maximal element, Zorn's lemma, union, chain, disjoint, collection, contains, property, implies, prime, iff, ideal, prime ideals, converse, multiplicatively closed, subset, ring
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This is version 8 of $m$-system, born on 2007-08-18, modified 2008-05-24.
Object id is 9873, canonical name is MSystem.
Accessed 889 times total.

Classification:
AMS MSC: 13B30 (Commutative rings and algebras :: Ring extensions and related topics :: Quotients and localization)
 16U20 (Associative rings and algebras :: Conditions on elements :: Ore rings, multiplicative sets, Ore localization)
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