Let R be a ring. A subset S of R is said to be an n-system if

  • S, and

  • for every xS, there is an rR, such that xrxS.

n-systems are a generalizationPlanetmathPlanetmath of m-systems (http://planetmath.org/MSystem) in a ring. Every m-system is an n-system, but not conversely. For example, for any distinct x,yR, inductively define the elements

a0=x, and ai+1=aiyiai for i=0,1,2,.

Form the set A={ann is a non-negative integer}. In additionPlanetmathPlanetmath, inductively define

b0=y, and bj+1=bjxjbj for j=0,1,2,

and form B={bmm is a non-negative integer}. Then both A and B are m-systems (as well as n-systems). Furthermore, S=AB is an n-system which is not an m-system.

The example above suggests that, given an n-system S and any xS, we can “construct” an m-system TS such that xT. Start with a0=x, inductively define ai+1=aiyiai, where the existence of yiR such that ai+1S is guaranteed by the fact that S is an n-system. Then the collectionMathworldPlanetmath T:={aii is a non-negative integer} is a subset of S that is an m-system. For if we pick any ai and aj, if ij, then ai is both the left and right sections of aj, meaning that there are r,sR such that aj=rai=ais (this can be easily proved inductively). As a result, ai(syj)aj=ajyjajS, and aj(yjr)ai=ajyjajS.

Remark n-systems provide another characterizationMathworldPlanetmath of a semiprime idealMathworldPlanetmath: an ideal IR is semiprime iff R-I is an n-system.


Suppose I is semiprime. Let xR-I. Then xRxI, which means there is an element yR such that xyxI. So R-I is an n-system. Now suppose that R-I is an n-system. Let xR with the condition that xRxI. This means xyxI for all yR. If xR-I, then there is some yR with xyxR-I, contradicting condition on x. Therefore, xI, and I is semiprime. ∎

Title n-system
Canonical name Nsystem
Date of creation 2013-03-22 17:29:29
Last modified on 2013-03-22 17:29:29
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 8
Author CWoo (3771)
Entry type Definition
Classification msc 13B30
Classification msc 16U20
Synonym n-system
Related topic MSystem
Related topic SemiprimeIdeal