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Difference between revisions of "Prime ideal"

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A two-sided [[Ideal|ideal]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074510/p0745101.png" /> of a [[Ring|ring]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074510/p0745102.png" /> such that the inclusion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074510/p0745103.png" /> for any two-sided ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074510/p0745104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074510/p0745105.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074510/p0745106.png" /> implies that either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074510/p0745107.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074510/p0745108.png" />. An ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074510/p0745109.png" /> of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074510/p07451010.png" /> is prime if and only if the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074510/p07451011.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074510/p07451013.png" />-system, i.e. for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074510/p07451014.png" /> there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074510/p07451015.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074510/p07451016.png" />. An ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074510/p07451017.png" /> of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074510/p07451018.png" /> is prime if and only if the quotient ring by it is a [[Prime ring|prime ring]].
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A two-sided [[Ideal|ideal]] $I$ of a [[Ring|ring]] $A$ such that the inclusion $PQ\subseteq I$ for any two-sided ideals $P$ and $Q$ of $A$ implies that either $P\subseteq I$ or $Q\subseteq I$. An ideal $I$ of a ring $R$ is prime if and only if the set $R\setminus I$ is an $m$-system, i.e. for any $a,b\in R\setminus I$ there exists an $x\in R$ such that $axb\in R\setminus I$. An ideal $I$ of a ring $A$ is prime if and only if the quotient ring by it is a [[Prime ring|prime ring]].
  
  
  
 
====Comments====
 
====Comments====
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This assumes that the empty set is an m system by default.
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This assumes that the empty set is an $m$-system by default.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.H. Rowen,  "Ring theory" , '''I''' , Acad. Press  (1988)  pp. 163</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.H. Rowen,  "Ring theory" , '''I''' , Acad. Press  (1988)  pp. 163</TD></TR></table>

Latest revision as of 10:46, 18 September 2016

A two-sided ideal $I$ of a ring $A$ such that the inclusion $PQ\subseteq I$ for any two-sided ideals $P$ and $Q$ of $A$ implies that either $P\subseteq I$ or $Q\subseteq I$. An ideal $I$ of a ring $R$ is prime if and only if the set $R\setminus I$ is an $m$-system, i.e. for any $a,b\in R\setminus I$ there exists an $x\in R$ such that $axb\in R\setminus I$. An ideal $I$ of a ring $A$ is prime if and only if the quotient ring by it is a prime ring.


Comments

This assumes that the empty set is an $m$-system by default.

References

[a1] L.H. Rowen, "Ring theory" , I , Acad. Press (1988) pp. 163
How to Cite This Entry:
Prime ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Prime_ideal&oldid=29171
This article was adapted from an original article by K.A. Zhevlakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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