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'$m$-system'
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| Title of object: |
$m$-system |
| Canonical Name: |
MSystem |
| Type: |
Definition |
| Created on: |
2007-08-18 13:55:33 |
| Modified on: |
2007-08-18 15:22:50 |
| Classification: |
msc:13B30, msc:16U20 |
| Synonyms: |
$m$-system=m-system |
Preamble:
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Content:
\PMlinkescapeword{closed subset}
Let $R$ be a ring. A subset $S$ of $R$ is called an $m$-{\em system} if
\begin{itemize}
\item $S\ne \varnothing$, and
\item for every two elements $x,\,y\in S$, there is an element $r\in R$ such that $xry\in S$.
\end{itemize}
$m$-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 $$\lbrace r^n\mid r\in R \mbox{ and } n \mbox{ is an odd positive integer}\rbrace$$ is an $m$-system, but not multiplicatively closed.
\textbf{Remarks}. $m$-Systems and prime ideals of a ring are intimately related. Two basic relationships between the two notions are
\begin{enumerate}
\item An ideal $P$ in a ring is a prime ideal iff $R\!\smallsetminus\!P$ is an $m$-system.
\item Given an $m$-system $S$ of $R$, there exists a prime ideal $P\subseteq R$ with the property that
$P\cap S = \varnothing$ and $P$ is the largest among all ideals with this property.
\end{enumerate}
$m$-Systems are also used to define the non-commutative version of the radical of an ideal of a ring. |
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