Nilpotent element

An element $ a $ of a ring or semi-group with zero $ A $ such that $ a ^ {n} = 0 $ for some natural number $ n $. The smallest such $ n $ is called the nilpotency index of $ a $. For example, in the residue ring modulo $ p ^ {n} $ (under multiplication), where $ p $ is a prime number, the residue class of $ p $ is nilpotent of index $ n $; in the ring of $ ( 2 \times 2 ) $-matrices with coefficients in a field $ K $ the matrix

$$ \left \| \begin{array}{cc} 0 & 1 \\ 0 & 0 \\ \end{array} \right \| $$

is nilpotent of index 2; in the group algebra $ F _ {p} [ G ] $, where $ F _ {p} $ is the field with $ p $ elements and $ G $ the cyclic group of order $ p $ generated by $ \sigma $, the element $ 1 - \sigma $ is nilpotent of index $ p $.

If $ a $ is a nilpotent element of index $ n $, then

$$ 1 = ( 1 - a ) ( 1 + a + \dots + a ^ {n- 1} ) , $$

that is, $ ( 1 - a ) $ is invertible in $ A $ and its inverse can be written as a polynomial in $ a $.

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

In the interpretation of a commutative ring $ A $ as the ring of functions on the space $ \mathop{\rm Spec} A $ (the spectrum of $ A $, 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 $ a $ of an associative ring $ R $ is strongly nilpotent if every sequence $ a= a _ {0} , a _ {1}, \dots $ such that $ a _ {n+ 1} \in a _ {n} R a _ {n} $ is ultimately zero. Obviously, every strongly-nilpotent element is nilpotent. The prime radical of a ring $ R $, 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