nilpotent

NILPOTENT ELEMENT

Definition 1 - 19
An element of a ring R is called nilpotent if there exists a positive
Integer n such that an =0 .
Example
Find the nilpotent of elements of Z9 .
Solution .
[3] 2 = [3 ] .9 [3] = [0] .
So nilpotent of [3] =2.
[6] 2 = [0]
So nilpotent of [6] = 2
Exercise
Prove that ; if R is an integral domain , then R dose not possess any
nonzero nilpotent element .
Solution
Let 0? a ? R
If an = 0 , so a . an-1 =0 , so an-1 = 0 ?
Continuing like this we obtain a = o .
Hence R dose not possess nonzero nilpotent element .

Exercises
If a and b are nilpotent elements of a commutative ring , show
that a + b is also nilpotent .

In 1 show that , if R is not commutative , then a + b may be not
nilpotent .