Nilpotents and Zero Divisors
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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
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Solution Summary
Nilpotents and Zero Divisors are investigated.
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Proof:
a) If x is not 0, since x^m=0, then x*(x^(m-1))=0. let y=x^(m-1), then xy=0. So x is a zero divisor. Thus either x is 0, or x is a zero divisor.
b) Since R is commutative, then (rx)^m=r^m * x^m =r^m * 0 =0. Thus rx is nilpotent for all r in ...
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