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A subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066670/n0666701.png" /> of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066670/n0666702.png" /> is called nil if each element of it is nilpotent (cf. [[Nilpotent element|Nilpotent element]]). An ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066670/n0666703.png" /> is a nil ideal if it is a nil subset. There is a largest nil ideal, which is called the nil radical. One has that
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066670/n0666705.png" /> denotes the [[Jacobson radical|Jacobson radical]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066670/n0666706.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066670/n0666707.png" /> is the prime radical of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066670/n0666708.png" />, i.e. the intersection of all prime ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066670/n0666709.png" />. Each of the inclusions can be proper. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066670/n06667010.png" /> is commutative, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066670/n06667011.png" />. The prime radical is also called the lower nil radical, and the nil radical the upper nil radical.
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A subset  $  A $
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of a ring  $  R $
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is called nil if each element of it is nilpotent (cf. [[Nilpotent element|Nilpotent element]]). An ideal of  $  R $
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is a nil ideal if it is a nil subset. There is a largest nil ideal, which is called the nil radical. One has that
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$$
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\mathop{\rm Jac} ( R)  \supset  \textrm{ Nil  Rad  } ( R)  \supset  \textrm{ Prime  Rad  } ( R),
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$$
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where  $  \mathop{\rm Jac} ( R) $
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denotes the [[Jacobson radical|Jacobson radical]] of $  R $
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and $  \textrm{ Prime  Rad  } ( R) $
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is the prime radical of $  R $,  
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i.e. the intersection of all prime ideals of $  R $.  
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Each of the inclusions can be proper. If $  R $
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is commutative, $  \textrm{ Nil  Rad  } ( R) = \textrm{ Prime  Rad  } ( R) $.  
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The prime radical is also called the lower nil radical, and the nil radical the upper nil radical.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Faith,  "Algebra" , '''II. Ring theory''' , Springer  (1976)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.C. McConnell,  J.C. Robson,  "Noncommutative Noetherian rings" , Wiley  (1987)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L.H. Rowen,  "Ring theory" , '''1''' , Acad. Press  (1988)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Faith,  "Algebra" , '''II. Ring theory''' , Springer  (1976)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.C. McConnell,  J.C. Robson,  "Noncommutative Noetherian rings" , Wiley  (1987)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L.H. Rowen,  "Ring theory" , '''1''' , Acad. Press  (1988)</TD></TR></table>

Latest revision as of 08:02, 6 June 2020


A subset $ A $ of a ring $ R $ is called nil if each element of it is nilpotent (cf. Nilpotent element). An ideal of $ R $ is a nil ideal if it is a nil subset. There is a largest nil ideal, which is called the nil radical. One has that

$$ \mathop{\rm Jac} ( R) \supset \textrm{ Nil Rad } ( R) \supset \textrm{ Prime Rad } ( R), $$

where $ \mathop{\rm Jac} ( R) $ denotes the Jacobson radical of $ R $ and $ \textrm{ Prime Rad } ( R) $ is the prime radical of $ R $, i.e. the intersection of all prime ideals of $ R $. Each of the inclusions can be proper. If $ R $ is commutative, $ \textrm{ Nil Rad } ( R) = \textrm{ Prime Rad } ( R) $. The prime radical is also called the lower nil radical, and the nil radical the upper nil radical.

References

[a1] C. Faith, "Algebra" , II. Ring theory , Springer (1976)
[a2] J.C. McConnell, J.C. Robson, "Noncommutative Noetherian rings" , Wiley (1987)
[a3] L.H. Rowen, "Ring theory" , 1 , Acad. Press (1988)
How to Cite This Entry:
Nil ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nil_ideal&oldid=47973
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