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A one- or two-sided ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066730/n0667301.png" /> in a ring or semi-group with zero such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066730/n0667302.png" /> for some natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066730/n0667303.png" />, that is, the product of any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066730/n0667304.png" /> elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066730/n0667305.png" /> vanishes. For example, in the residue class ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066730/n0667306.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066730/n0667307.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066730/n0667308.png" /> is a prime number, every ideal except the ring itself is nilpotent. In the group ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066730/n0667309.png" /> of a finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066730/n06673010.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066730/n06673011.png" /> over the field with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066730/n06673012.png" /> elements the ideal generated by the elements of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066730/n06673013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066730/n06673014.png" />, is nilpotent. In the ring of upper-triangular matrices over a field the matrices with 0's along the main diagonal form a nilpotent ideal.
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A one- or two-sided ideal $M$ in a ring or semi-group with zero such that $M^n=\{0\}$ for some natural number $n$, that is, the product of any $n$ elements of $M$ vanishes. For example, in the residue class ring $\mathbf Z/p^n\mathbf Z$ modulo $p^n$, where $p$ is a prime number, every ideal except the ring itself is nilpotent. In the group ring $\mathbf F_p[G]$ of a finite $p$-group $G$ over the field with $p$ elements the ideal generated by the elements of the form $\sigma-1$, $\sigma\in G$, is nilpotent. In the ring of upper-triangular matrices over a field the matrices with 0's along the main diagonal form a nilpotent ideal.
  
 
Every element of a nilpotent ideal is nilpotent. Every nilpotent ideal is also a nil ideal and is contained in the [[Jacobson radical|Jacobson radical]] of the ring. In Artinian rings the Jacobson radical is nilpotent, and the concepts of a nilpotent ideal and a nil ideal coincide. The latter property also holds in a [[Noetherian ring|Noetherian ring]]. In a left (or right) Noetherian ring every left (right) nil ideal is nilpotent.
 
Every element of a nilpotent ideal is nilpotent. Every nilpotent ideal is also a nil ideal and is contained in the [[Jacobson radical|Jacobson radical]] of the ring. In Artinian rings the Jacobson radical is nilpotent, and the concepts of a nilpotent ideal and a nil ideal coincide. The latter property also holds in a [[Noetherian ring|Noetherian ring]]. In a left (or right) Noetherian ring every left (right) nil ideal is nilpotent.
  
All nilpotent ideals of a commutative ring are contained in the nil radical, which, in general, need not be a nilpotent but only a nil ideal. A simple example of this situation is the direct sum of the rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066730/n06673015.png" /> for all natural numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066730/n06673016.png" />. In a commutative ring every [[Nilpotent element|nilpotent element]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066730/n06673017.png" /> is contained in some nilpotent ideal, for example, in the [[Principal ideal|principal ideal]] generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066730/n06673018.png" />. In a non-commutative ring there may by nilpotent elements that are not contained in any nilpotent ideal (nor even in a nil ideal). For example, in the general matrix ring over a field there are nilpotent elements; in particular, the nilpotent matrices mentioned above, in which the only non-zero elements stand above the main diagonal, but since the ring is simple, it has no non-zero nilpotent ideals.
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All nilpotent ideals of a commutative ring are contained in the nil radical, which, in general, need not be a nilpotent but only a nil ideal. A simple example of this situation is the direct sum of the rings $\mathbf Z/p^n\mathbf Z$ for all natural numbers $n$. In a commutative ring every [[Nilpotent element|nilpotent element]] $a$ is contained in some nilpotent ideal, for example, in the [[Principal ideal|principal ideal]] generated by $a$. In a non-commutative ring there may by nilpotent elements that are not contained in any nilpotent ideal (nor even in a nil ideal). For example, in the general matrix ring over a field there are nilpotent elements; in particular, the nilpotent matrices mentioned above, in which the only non-zero elements stand above the main diagonal, but since the ring is simple, it has no non-zero nilpotent ideals.
  
In a finite-dimensional [[Lie algebra|Lie algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066730/n06673019.png" /> there is maximal nilpotent ideal, which consists of the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066730/n06673020.png" /> for which the endomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066730/n06673021.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066730/n06673022.png" /> is nilpotent.
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In a finite-dimensional [[Lie algebra|Lie algebra]] $G$ there is maximal nilpotent ideal, which consists of the elements $x\in G$ for which the endomorphism $y\to[x,y]$ for $y\in G$ is nilpotent.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Lang,  "Algebra" , Addison-Wesley  (1974)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Jacobson,  "Structure of rings" , Amer. Math. Soc.  (1956)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Faith,  "Algebra: rings, modules, and categories" , '''1''' , Springer  (1973)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.N. Herstein,  "Noncommutative rings" , Math. Assoc. Amer.  (1968)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley  (1975)  (Translated from French)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Lang,  "Algebra" , Addison-Wesley  (1974)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Jacobson,  "Structure of rings" , Amer. Math. Soc.  (1956)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Faith,  "Algebra: rings, modules, and categories" , '''1''' , Springer  (1973)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.N. Herstein,  "Noncommutative rings" , Math. Assoc. Amer.  (1968)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley  (1975)  (Translated from French)</TD></TR></table>

Latest revision as of 23:58, 24 November 2018

A one- or two-sided ideal $M$ in a ring or semi-group with zero such that $M^n=\{0\}$ for some natural number $n$, that is, the product of any $n$ elements of $M$ vanishes. For example, in the residue class ring $\mathbf Z/p^n\mathbf Z$ modulo $p^n$, where $p$ is a prime number, every ideal except the ring itself is nilpotent. In the group ring $\mathbf F_p[G]$ of a finite $p$-group $G$ over the field with $p$ elements the ideal generated by the elements of the form $\sigma-1$, $\sigma\in G$, is nilpotent. In the ring of upper-triangular matrices over a field the matrices with 0's along the main diagonal form a nilpotent ideal.

Every element of a nilpotent ideal is nilpotent. Every nilpotent ideal is also a nil ideal and is contained in the Jacobson radical of the ring. In Artinian rings the Jacobson radical is nilpotent, and the concepts of a nilpotent ideal and a nil ideal coincide. The latter property also holds in a Noetherian ring. In a left (or right) Noetherian ring every left (right) nil ideal is nilpotent.

All nilpotent ideals of a commutative ring are contained in the nil radical, which, in general, need not be a nilpotent but only a nil ideal. A simple example of this situation is the direct sum of the rings $\mathbf Z/p^n\mathbf Z$ for all natural numbers $n$. In a commutative ring every nilpotent element $a$ is contained in some nilpotent ideal, for example, in the principal ideal generated by $a$. In a non-commutative ring there may by nilpotent elements that are not contained in any nilpotent ideal (nor even in a nil ideal). For example, in the general matrix ring over a field there are nilpotent elements; in particular, the nilpotent matrices mentioned above, in which the only non-zero elements stand above the main diagonal, but since the ring is simple, it has no non-zero nilpotent ideals.

In a finite-dimensional Lie algebra $G$ there is maximal nilpotent ideal, which consists of the elements $x\in G$ for which the endomorphism $y\to[x,y]$ for $y\in G$ is nilpotent.

References

[1] S. Lang, "Algebra" , Addison-Wesley (1974)
[2] N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956)
[3] C. Faith, "Algebra: rings, modules, and categories" , 1 , Springer (1973)
[4] I.N. Herstein, "Noncommutative rings" , Math. Assoc. Amer. (1968)
[5] N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French)
How to Cite This Entry:
Nilpotent ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nilpotent_ideal&oldid=18365
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article