Question

An element of a commutative ring R(1≠0) is nilpotent if __________

(a) a+1=0

(b) a^n = 0, for some positive integer n

(c) a^n = 1, for some integer n

(d) a^2 = 0

This question was addressed to me by my college director while I was bunking the class.

My question is based upon Cyclic Groups in division Groups of Discrete Mathematics

Answer 1

The correct choice is (b) a^n = 0, for some positive integer n

Explanation: Since a is nilpotent in a commutative ring R, we have an=0 for some positive integer n. since R is commutative, for any m∈R, we have (am)n=anmn=0. Then we have the following equality: (1−am)(1+(am)+(am)2+⋯+(am)n−1)=1. Hence, 1−am is a unit in R.