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et R be a PID. 5) Prove that every nonzero nonunit element is divisible by a prime lement. i) If {In}neN is a sequence of ideals of R such that I₁ C 1₂ C C In C .., prove that there exists a positive integer n such that In - ... = Intl ii) Prove that every nonzero nonunit can be expressed as a finite product f prime elements.

et R be a PID. 5) Prove that every nonzero nonunit element is divisible by a prime lement. i) If {In}neN is a sequence of ideals of R such that I₁ C 1₂ C C In C .., prove that there exists a positive integer n such that In - ... = Intl ii) Prove that every nonzero nonunit can be expressed as a finite product f prime elements.

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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Anstract Albegra:

Maximal, Prime, and Primary Ideals 

Let R be a PID. (i) Prove that every nonzero nonunit element is divisible by a prime element. (ii) If {In}neN is a sequence of ideals of R such that I₁ C 1₂ C... C In C prove that there exists a positive integer n such that In = Intl , (iii) Prove that every nonzero nonunit can be expressed as a finite product of prime elements.
Transcribed Image Text:Let R be a PID. (i) Prove that every nonzero nonunit element is divisible by a prime element. (ii) If {In}neN is a sequence of ideals of R such that I₁ C 1₂ C... C In C prove that there exists a positive integer n such that In = Intl , (iii) Prove that every nonzero nonunit can be expressed as a finite product of prime elements.
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