Nile potent element

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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 .


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.


For example, the matrix is
nilpotent, because it applies
(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 .
  • In the residual class ring , the nilpotent elements are exactly the residual classes of 0 and 6.
  • The zero element of a ring is always nilpotent, there is.


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 .

  • Is , then is and is zero divisor , because and .

If there is also a ring with 1 and not the zero ring , then:

  • is not invertible (with regard to the multiplication), because the contradiction follows for a ring element ( was chosen minimally!).
  • is invertible because it applies .
  • If there is a unit of that also commutes, then what one sees as by looking at the representation can also be inverted .

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 .

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.