Let x be a nilpotent element of the commutative ring r i.e., Ask an Expert

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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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