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- Jun 22, 2012

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Dummit and Foote, Section 8.2 (Principal Ideal Domains (PIDs) ) - Exercise 4, page 282.

Let R be an integral domain.

Prove that if the following two conditions hold then R is a Principal Ideal Domain:

(i) any two non-zero elements a and b in R have a greatest common divisor which can be written in the form ra + sb for some [TEX] r, s \in R [/TEX] and

(ii) if [TEX] a_1, a_2, a_3, ... [/TEX] are non-zero elements of R such that [TEX] a_{i+1} | a_i [/TEX] for all i, then there is a positive integer N such that [TEX] a_n [/TEX] is a unit times [TEX] a_N [/TEX] for all [TEX] n \ge N [/TEX]

I am somewhat overwhelmed by this exercise. I would appreciate it if someone could help me get started and indicate a strategy for formulating a proof.

Peter

Let R be an integral domain.

Prove that if the following two conditions hold then R is a Principal Ideal Domain:

(i) any two non-zero elements a and b in R have a greatest common divisor which can be written in the form ra + sb for some [TEX] r, s \in R [/TEX] and

(ii) if [TEX] a_1, a_2, a_3, ... [/TEX] are non-zero elements of R such that [TEX] a_{i+1} | a_i [/TEX] for all i, then there is a positive integer N such that [TEX] a_n [/TEX] is a unit times [TEX] a_N [/TEX] for all [TEX] n \ge N [/TEX]

I am somewhat overwhelmed by this exercise. I would appreciate it if someone could help me get started and indicate a strategy for formulating a proof.

Peter

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