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1. Let R be a PID (principal ideal domain) and let a, b E R be non-zero, non- units. Recall: an element d is called a greatest common divisor (GCD) of a and b, denoted gcd(a, b), if d | a and d | b and if c | a and c | b then c | d. PROVE that if a, b E R are non-zero, non-units, then (a) gcd(a, b) exists and (b) gcd(a, b) = za +yb for some x, y E R (HINT: consier - the ideal generated by a and b)

1. Let R be a PID (principal ideal domain) and let a, b E R be non-zero, non- units. Recall: an element d is called a greatest common divisor (GCD) of a and b, denoted gcd(a, b), if d | a and d | b and if c | a and c | b then c | d. PROVE that if a, b E R are non-zero, non-units, then (a) gcd(a, b) exists and (b) gcd(a, b) = za +yb for some x, y E R (HINT: consier - the ideal generated by a and b)

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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1. Let R be a PID (principal ideal domain) and let a, b E R be non-zero, non- units. Recall: an element d is called a greatest common divisor (GCD) of a and b, denoted gcd(a, b), if d | a and d | b and if c | a and c | b then c | d. PROVE that if a, b E R are non-zero, non-units, then (a) god(a, b) exists and (b) gcd(a, b) = xa+yb for some x, y E R (HINT: consier <a, b> - the ideal generated by a and b)
Transcribed Image Text:1. Let R be a PID (principal ideal domain) and let a, b E R be non-zero, non- units. Recall: an element d is called a greatest common divisor (GCD) of a and b, denoted gcd(a, b), if d | a and d | b and if c | a and c | b then c | d. PROVE that if a, b E R are non-zero, non-units, then (a) god(a, b) exists and (b) gcd(a, b) = xa+yb for some x, y E R (HINT: consier <a, b> - the ideal generated by a and b)
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