# Nile potent element

A nilpotent element is a term from ring theory , a branch of mathematics . An element of a ring is called nilpotent if it is multiplied enough often by itself to produce the zero element .

## definition

An element of a ring is called nilpotent if there is a positive natural number such that . An ideal is said to be nilpotent if there is a positive natural number such that holds. ${\ displaystyle x}$ ${\ displaystyle R}$ ${\ displaystyle n}$${\ displaystyle x ^ {n} = 0}$ ${\ displaystyle I \ subseteq R}$${\ displaystyle n}$${\ displaystyle I ^ {n} = (0)}$

## Examples

For example, the matrix is
${\ displaystyle A = {\ begin {pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \ end {pmatrix}}}$
nilpotent, because it applies
${\ displaystyle A ^ {3} = 0}$.
(For special properties of nilpotent matrices see the article nilpotent matrix .)
• In the remainder class ring , the remainder classes of 0, 2, 4 and 6 are nilpotent, since their third power is congruent to 0 modulo 8. In this ring, each element is either nilpotent or a unity .${\ displaystyle \ mathbb {Z} / 8 \ mathbb {Z}}$
• In the residual class ring , the nilpotent elements are exactly the residual classes of 0 and 6.${\ displaystyle \ mathbb {Z} / 12 \ mathbb {Z}}$
• The zero element of a ring is always nilpotent, there is.${\ displaystyle 0 ^ {1} = 0}$

## properties

The set of all nilpotent elements of a commutative ring forms an ideal, the so-called Nile radical .

The intersection of all prime ideals in a commutative ring with 1 is exactly the Nile radical.

In the following be a ring, a nilpotent element of and the smallest natural number with . ${\ displaystyle R}$${\ displaystyle a}$${\ displaystyle R}$${\ displaystyle n}$${\ displaystyle a ^ {n} = 0}$

• Is , then is and is zero divisor , because and .${\ displaystyle a \ neq 0}$${\ displaystyle n> 1}$${\ displaystyle a}$${\ displaystyle aa ^ {n-1} = 0}$${\ displaystyle a ^ {n-1} \ neq 0}$

If there is also a ring with 1 and not the zero ring , then: ${\ displaystyle R}$

• ${\ displaystyle a}$is not invertible (with regard to the multiplication), because the contradiction follows for a ring element ( was chosen minimally!).${\ displaystyle from = 1}$${\ displaystyle b}$${\ displaystyle 0 = a ^ {n} b = a ^ {n-1}}$${\ displaystyle n}$
• ${\ displaystyle 1-a}$is invertible because it applies .${\ displaystyle (1-a) \ left (1 + a + a ^ {2} + \ dotsb + a ^ {n-1} \ right) = 1-a ^ {n} = 1 = \ left (1+ a + a ^ {2} + \ dotsb + a ^ {n-1} \ right) (1-a)}$
• If there is a unit of that also commutes, then what one sees as by looking at the representation can also be inverted .${\ displaystyle b}$${\ displaystyle R}$${\ displaystyle a}$${\ displaystyle b + a}$${\ displaystyle b + a = b \ cdot \ left (1- \ left (-b ^ {- 1} a \ right) \ right)}$

Let be a residue class ring and the product of all prime divisors of , ie all prime numbers that appear in the prime factorization of . Eg for is . Then the nilpotent elements of are exactly the remainder classes of integers that are multiples of . The idea of ​​the proof is as follows: If the largest exponent that occurs in the prime factorization of , then is a multiple of ; every number for which a power is a multiple of must itself have every prime divisor of . ${\ displaystyle R}$${\ displaystyle \ mathbb {Z} / m \ mathbb {Z}}$${\ displaystyle p}$${\ displaystyle m}$${\ displaystyle m}$${\ displaystyle m = 12 = 2 ^ {2} \ cdot 3}$${\ displaystyle p = 6 = 2 \ cdot 3}$${\ displaystyle R}$${\ displaystyle p}$${\ displaystyle k}$${\ displaystyle m}$${\ displaystyle p ^ {k}}$${\ displaystyle m}$${\ displaystyle m}$${\ displaystyle m}$

A ring that contains no nilpotent elements other than zero is called reduced .

## Individual evidence

1. ^ Serge Lang : Algebra , 3rd edition, Graduate Texts in Mathematics, Springer Verlag 2005, ISBN 978-0387953854 , p. 417.