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}$

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