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

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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