An element of a commutative ring R(1≠0) is nilpotent if __________
A. a+1=0
B. an= 0, for some positive integer n
C. an= 1, for some integer n
D. a2 = 0
Answer:B
An element of a commutative ring R(1≠0) is nilpotent if __________
A. a+1=0 B. an= 0, for some positive integer n C. an= 1, for some integer n D. a2 = 0
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.