Nile ideal

Nilideal is a mathematical term from ring theory .

definition

Let R be a ring . An ideal N of R that consists only of nilpotent elements is called a Nile ideal .

More generally, each subset of a ring is called nil if it only consists of nilpotent elements.

While one demands of a nilpotent ideal that there is a with , that is, every product of the length of elements is equal to 0, a Nile ideal is only required that for every element there is a dependent with . ${\displaystyle I\subset R}$ ${\displaystyle n}$ ${\displaystyle I^{n}=\{0\}}$ ${\displaystyle a_{1}\cdot \ldots \cdot a_{n}}$ ${\displaystyle n}$ ${\displaystyle a_{i}\in I}$ ${\displaystyle a\in I}$ ${\displaystyle a}$ ${\displaystyle n}$ ${\displaystyle a^{n}=0}$

Examples and characteristics

• Every nilpotent ideal is a Nile ideal, and for finitely generated ideals in commutative rings the converse also applies. An example of a Nile ideal that is not nilpotent is the ideal in the ring with one body and one indefinite for each natural number . ${\displaystyle (X_{1},X_{2},\ldots )}$ ${\displaystyle k[X_{1},X_{2},\ldots ]/(X_{1},X_{2}^{2},X_{3}^{3},\ldots )}$ ${\displaystyle k}$ ${\displaystyle X_{i}}$ ${\displaystyle i}$

• According to one of Levitzki's theorem , every left or right Nile ideal in a left Noetherian ring is already nilpotent.

• The prime radical is a Nile ideal.

Individual evidence

1. ↑ Louis H. Rowen: Ring Theory. Volume 1. Academic Press Inc., Boston et al. 1988, ISBN 0-125-99841-4 ( Pure and Applied Mathematics 127), page 41
2. ↑ Louis H. Rowen: Ring Theory. Volume 1. Academic Press Inc., Boston et al. 1988, ISBN 0-125-99841-4 ( Pure and Applied Mathematics 127), sentence 2.6.23
3. ↑ Louis H. Rowen: Ring Theory. Volume 1. Academic Press Inc., Boston et al. 1988, ISBN 0-125-99841-4 ( Pure and Applied Mathematics 127), sentence 2.6.15