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An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n0667101.png" /> of a ring or semi-group with zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n0667102.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n0667103.png" /> for some natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n0667104.png" />. The smallest such <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n0667105.png" /> is called the nilpotency index of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n0667106.png" />. For example, in the residue ring modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n0667107.png" /> (under multiplication), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n0667108.png" /> is a prime number, the residue class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n0667109.png" /> is nilpotent of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n06671010.png" />; in the ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n06671011.png" />-matrices with coefficients in a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n06671012.png" /> the matrix
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n0667101.png
 
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$#A+1 = 40 n = 0
 
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$#C+1 = 40 : ~/encyclopedia/old_files/data/N066/N.0606710 Nilpotent element
 
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n06671013.png" /></td> </tr></table>
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An element  $  a $
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is nilpotent of index 2; in the group algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n06671014.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n06671015.png" /> is the field with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n06671016.png" /> elements and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n06671017.png" /> the cyclic group of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n06671018.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n06671019.png" />, the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n06671020.png" /> is nilpotent of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n06671021.png" />.
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of a ring or semi-group with zero  $  A $
 
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such that  $  a  ^ {n} = 0 $
 
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for some natural number  $  n $.  
 
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The smallest such  $  n $
 
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is called the nilpotency index of $  a $.  
 
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For example, in the residue ring modulo  $  p  ^ {n} $(
 
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under multiplication), where  $  p $
 
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is a prime number, the residue class of  $  p $
 
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is nilpotent of index $  n $;
 
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in the ring of  $  ( 2 \times 2 ) $-
 
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matrices with coefficients in a field  $  K $
 
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the matrix
 
  
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$$
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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n06671022.png" /> is a nilpotent element of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n06671023.png" />, then
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\left \|
 
  
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is nilpotent of index 2; in the group algebra  $  F _ {p} [ G ] $,
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n06671024.png" /></td> </tr></table>
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where  $  F _ {p} $
 
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is the field with  $  p $
 
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elements and  $  G $
 
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the cyclic group of order  $  p $
 
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generated by  $  \sigma $,
 
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the element  $  1 - \sigma $
 
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is nilpotent of index  $  p $.
 
  
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If  $  a $
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that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n06671025.png" /> is invertible in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n06671026.png" /> and its inverse can be written as a polynomial in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n06671027.png" />.
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is a nilpotent element of index  $  n $,
 
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then
 
  
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$$
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In a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n06671028.png" /> an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n06671029.png" /> is nilpotent if and only if it is contained in all prime ideals of the ring. All nilpotent elements form an ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n06671030.png" />, the so-called nil radical of the ring; it coincides with the intersection of all prime ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n06671031.png" />. The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n06671032.png" /> has no non-zero nilpotent elements.
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1  = ( 1 - a ) ( 1 + a + \dots + a  ^ {n-} 1 ) ,
 
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$$
 
  
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that is, $  ( 1 - a ) $
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In the interpretation of a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n06671033.png" /> as the ring of functions on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n06671034.png" /> (the spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n06671035.png" />, cf. [[Spectrum of a ring|Spectrum of a ring]]), the nilpotent elements correspond to functions that vanish identically. Nevertheless, the consideration of nilpotent elements frequently turns out to be useful in algebraic geometry because it makes it possible to obtain purely algebraic analogues of a number of concepts in analysis and differential geometry (infinitesimal deformations, etc.).
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is invertible in $  A $
 
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and its inverse can be written as a polynomial in $  a $.
 
  
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In a commutative ring  $  A $
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====References====
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an element  $  a $
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S. Lang, "Algebra" , Addison-Wesley (1974) {{MR|0783636}} {{ZBL|0712.00001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> I.N. Herstein, "Noncommutative rings" , Math. Assoc. Amer. (1968) {{MR|1535024}} {{MR|0227205}} {{ZBL|0177.05801}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR></table>
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is nilpotent if and only if it is contained in all prime ideals of the ring. All nilpotent elements form an ideal  $  J $,  
 
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the so-called nil radical of the ring; it coincides with the intersection of all prime ideals of  $  A $.  
 
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The ring  $  A / J $
 
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has no non-zero nilpotent elements.
 
  
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In the interpretation of a commutative ring  $  A $
 
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as the ring of functions on the space  $  \mathop{\rm Spec}  A $(
 
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the spectrum of  $  A $,
 
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cf. [[Spectrum of a ring|Spectrum of a ring]]), the nilpotent elements correspond to functions that vanish identically. Nevertheless, the consideration of nilpotent elements frequently turns out to be useful in algebraic geometry because it makes it possible to obtain purely algebraic analogues of a number of concepts in analysis and differential geometry (infinitesimal deformations, etc.).
 
  
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====References====
 
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S. Lang, "Algebra" , Addison-Wesley (1974) {{MR|0783636}} {{ZBL|0712.00001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> I.N. Herstein, "Noncommutative rings" , Math. Assoc. Amer. (1968) {{MR|1535024}} {{MR|0227205}} {{ZBL|0177.05801}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR></table>
 
  
 
====Comments====
 
====Comments====
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An element $  a $
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An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n06671036.png" /> of an associative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n06671037.png" /> is strongly nilpotent if every sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n06671038.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n06671039.png" /> is ultimately zero. Obviously, every strongly-nilpotent element is nilpotent. The prime radical of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066710/n06671040.png" />, i.e. the intersection of all prime ideals, consists of precisely the strongly-nilpotent elements. It is a [[Nil ideal|nil ideal]].
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of an associative ring $  R $
 
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is strongly nilpotent if every sequence $  a= a _ {0} , a _ {1} \dots $
 
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such that $  a _ {n+} 1 \in a _ {n} R a _ {n} $
 
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is ultimately zero. Obviously, every strongly-nilpotent element is nilpotent. The prime radical of a ring $  R $,  
 
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i.e. the intersection of all prime ideals, consists of precisely the strongly-nilpotent elements. It is a [[Nil ideal|nil ideal]].
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.C. McConnell, J.C. Robson, "Noncommutative Noetherian rings" , Wiley (1987) pp. §0.2 {{MR|934572}} {{ZBL|}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.C. McConnell, J.C. Robson, "Noncommutative Noetherian rings" , Wiley (1987) pp. §0.2 {{MR|934572}} {{ZBL|}} </TD></TR></table>

Revision as of 14:52, 7 June 2020

An element of a ring or semi-group with zero such that for some natural number . The smallest such is called the nilpotency index of . For example, in the residue ring modulo (under multiplication), where is a prime number, the residue class of is nilpotent of index ; in the ring of -matrices with coefficients in a field the matrix

is nilpotent of index 2; in the group algebra , where is the field with elements and the cyclic group of order generated by , the element is nilpotent of index .

If is a nilpotent element of index , then

that is, is invertible in and its inverse can be written as a polynomial in .

In a commutative ring an element is nilpotent if and only if it is contained in all prime ideals of the ring. All nilpotent elements form an ideal , the so-called nil radical of the ring; it coincides with the intersection of all prime ideals of . The ring has no non-zero nilpotent elements.

In the interpretation of a commutative ring as the ring of functions on the space (the spectrum of , cf. Spectrum of a ring), the nilpotent elements correspond to functions that vanish identically. Nevertheless, the consideration of nilpotent elements frequently turns out to be useful in algebraic geometry because it makes it possible to obtain purely algebraic analogues of a number of concepts in analysis and differential geometry (infinitesimal deformations, etc.).

References

[1] S. Lang, "Algebra" , Addison-Wesley (1974) MR0783636 Zbl 0712.00001
[2] I.N. Herstein, "Noncommutative rings" , Math. Assoc. Amer. (1968) MR1535024 MR0227205 Zbl 0177.05801
[3] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001


Comments

An element of an associative ring is strongly nilpotent if every sequence such that is ultimately zero. Obviously, every strongly-nilpotent element is nilpotent. The prime radical of a ring , i.e. the intersection of all prime ideals, consists of precisely the strongly-nilpotent elements. It is a nil ideal.

References

[a1] J.C. McConnell, J.C. Robson, "Noncommutative Noetherian rings" , Wiley (1987) pp. §0.2 MR934572
How to Cite This Entry:
Nilpotent element. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nilpotent_element&oldid=49329
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
Retrieved from "https://encyclopediaofmath.org/index.php?title=Nilpotent_element&oldid=49329"