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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.

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.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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The question is in the attached image, if able focus on the way a formal proof is written, write the proof in a way that its like giving a lecture about this question. Thank you in advance.

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.
Transcribed Image Text: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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