Nilideal

Nil ideal is a mathematical term used in ring theory.

Definition

Let R be a ring. An ideal N of R, which consists only of nilpotent elements, ie nil ideal.

General called every subset of a ring nil if it consists only of nilpotent elements.

While one of a nilpotent ideal requires that there is a with, that is, each product of the length of elements is equal to 0, it is only required of a nil ideal that to each element of a dependent are with.

Examples and properties

• Each nilpotent ideal is a nil ideal, and for finitely generated ideals in commutative rings converse is also true. An example of a nil ideal, which is not nilpotent, is the ideal in the ring with a body and each one indeterminate for each natural number.

• By a theorem of Levitzki every left or right nil ideal is nilpotent already in a left - noetherian ring.

• The Primradikal is a nil ideal.