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

  • S, and

  • for every two elements x,yS, there is an element rR such that xryS.

m-Systems are a generalizationPlanetmathPlanetmath of multiplicatively closet subsets in a ring. Indeed, every multiplicatively closed subset of R is an m-system: any x,yS, then xyS, hence xyyS. However, the converseMathworldPlanetmath is not true. For example, the set

{rnrR and n 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 idealsMathworldPlanetmathPlanetmathPlanetmath of a ring are intimately related. Two basic relationships between the two notions are

  1. 1.

    An ideal P in a ring R is a prime ideal iff R-P is an m-system.


    P is prime iff xRyP implies x or yP, iff x,yR-P implies that there is rR with xryP iff R-P is an m-system. ∎

  2. 2.

    Given an m-system S of R and an ideal I with IS=. Then there exists a prime ideal PR with the property that P contains I and PS=, and P is the largest among all ideals with this property.


    Let 𝒞 be the collectionMathworldPlanetmath of all ideals containing I and disjoint from S. First, I𝒞. Second, any chain K of ideals in 𝒞, its union K is also in 𝒞. So Zorn’s lemma applies. Let P be a maximal elementMathworldPlanetmath in 𝒞. We want to show that P is prime. Suppose otherwise. In other words, aRbP with a,bP. Then P,a and P,b both have non-empty intersectionsMathworldPlanetmath with S. Let

    c=p+fagP,aS and d=q+hbkP,bS,

    where p,qP and f,g,h,kR. Then there is rR such that crdS. But this implies that


    as well, contradicting PS=. Therefore, P is prime. ∎

m-Systems are also used to define the non-commutative version of the radicalPlanetmathPlanetmathPlanetmathPlanetmath of an ideal of a ring.

Title m-system
Canonical name Msystem
Date of creation 2013-03-22 17:29:09
Last modified on 2013-03-22 17:29:09
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 11
Author CWoo (3771)
Entry type Definition
Classification msc 16U20
Classification msc 13B30
Synonym m-system
Related topic MultiplicativelyClosed
Related topic NSystem
Related topic PrimeIdeal