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 = 0This 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

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

1.	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 = 0This 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» 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.

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