An element x in R is called nilpotent if x = 0 for some m € 2¹. Let x be a nilpotent element of the commutative ring R (a) Prove that x is either zero or a zero divisor. (b) Prove that rx is nilpotent for all r € 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.

An element x in R is called nilpotent if x = 0 for some m € 2¹. Let x be a nilpotent element of the commutative ring R (a) Prove that x is either zero or a zero divisor. (b) Prove that rx is nilpotent for all r € 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.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.1: Definition Of A Ring

Problem 49E: An element a of a ring R is called nilpotent if an=0 for some positive integer n. Prove that the set...

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please provide some explanation with the taken steps in the attached exercise, Im quite new to abstract algebra

An element x in R is called nilpotent if x = 0 for some m € Z+.
Let x be a nilpotent element of the commutative ring R
(a) Prove that x is either zero or a zero divisor.
(b) Prove that rx is nilpotent for all r & 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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