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

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$S\ne \mathrm{\varnothing}$, and

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for every $x\in S$, there is an $r\in R$, such that $xrx\in S$.
$n$systems are a generalization^{} 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 $x,y\in R$, inductively define the elements
$${a}_{0}=x,\text{and}{a}_{i+1}={a}_{i}{y}^{i}{a}_{i}\mathit{\hspace{1em}}\text{for}i=0,1,2,\mathrm{\dots}.$$ 
Form the set $A=\{{a}_{n}\mid n\text{is a nonnegative integer}\}$. In addition^{}, inductively define
$${b}_{0}=y,\text{and}{b}_{j+1}={b}_{j}{x}^{j}{b}_{j}\mathit{\hspace{1em}}\text{for}j=0,1,2\mathrm{\dots},$$ 
and form $B=\{{b}_{m}\mid m\text{is a nonnegative integer}\}$. Then both $A$ and $B$ are $m$systems (as well as $n$systems). Furthermore, $S=A\cup B$ is an $n$system which is not an $m$system.
The example above suggests that, given an $n$system $S$ and any $x\in S$, we can “construct” an $m$system $T\subseteq S$ such that $x\in T$. Start with ${a}_{0}=x$, inductively define ${a}_{i+1}={a}_{i}{y}_{i}{a}_{i}$, where the existence of ${y}_{i}\in R$ such that ${a}_{i+1}\in S$ is guaranteed by the fact that $S$ is an $n$system. Then the collection^{} $T:=\{{a}_{i}\mid i\text{is a nonnegative integer}\}$ is a subset of $S$ that is an $m$system. For if we pick any ${a}_{i}$ and ${a}_{j}$, if $i\le j$, then ${a}_{i}$ is both the left and right sections of ${a}_{j}$, meaning that there are $r,s\in R$ such that ${a}_{j}=r{a}_{i}={a}_{i}s$ (this can be easily proved inductively). As a result, ${a}_{i}(s{y}_{j}){a}_{j}={a}_{j}{y}_{j}{a}_{j}\in S$, and ${a}_{j}({y}_{j}r){a}_{i}={a}_{j}{y}_{j}{a}_{j}\in S$.
Remark $n$systems provide another characterization^{} of a semiprime ideal^{}: an ideal $I\subseteq R$ is semiprime iff $RI$ is an $n$system.
Proof.
Suppose $I$ is semiprime. Let $x\in RI$. Then $xRx\u2288I$, which means there is an element $y\in R$ such that $xyx\notin I$. So $RI$ is an $n$system. Now suppose that $RI$ is an $n$system. Let $x\in R$ with the condition that $xRx\subseteq I$. This means $xyx\in I$ for all $y\in R$. If $x\in RI$, then there is some $y\in R$ with $xyx\in RI$, contradicting condition on $x$. Therefore, $x\in I$, and $I$ is semiprime. ∎
Title  $n$system 

Canonical name  Nsystem 
Date of creation  20130322 17:29:29 
Last modified on  20130322 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  nsystem 
Related topic  MSystem 
Related topic  SemiprimeIdeal 