How to prove directly sum of non zero divisor and nilpotent is again non zero divisor?
I know that it can be easily proved by extending ring to ring of fraction So that I have a unit as that non zero divisor and then I can prove sum of unit and nilpotent is again unit.
But I was thinking to prove it by a more direct way?
My attempt in this regard:
a is non zero divisor. b is nilpotent of index n
On the contrary, suppose a+b is zero divisor so there is $c\neq 0$
Case 1: $c$ is not a nilpotent element of order $n$
multiplying each side n time
But $a$ is not zero divisor this implies $c$ is the nilpotent element of order $n$
CAse 2: If $c$ is the nilpotent element of order $n$
Then I am unable to find the contradiction.
Any Help will be appreciated