Assignment:
Let x be a nilpotent element of the commutative ring R i.e. x^m for some positive integer m
a) Prove that x is either zero or a zero divisor
b) Prove that rx is nilpotent for all r in R
c) Prove that 1+x is a unit in R
d) Deduce that the sum of a nilpotent element and a unit is a unit
Provide complete and step by step solution for the question and show calculations and use formulas.