Mathematics 893 Problem Set 7

Suggested reading: Chapter 8, sections 2-3, Chapter 9, sections 1 and 3. Chapter 10, sections 1-3.

Solutions due Monday November 14.

(a) Page 277 # 1(a), 1(d), 2(b), 4. Apply #4 to question 5(c).

(b) Page 282 #1, Page 298 #4, 6

(c) Page 306 #1, 2,

(d) Page 306 #4

(e) Page 343 #1, 8, 9 Remark: Remember that the text uses (a, b) for the greatest common divisor of a and b. However I prefer gcd(a, b) for the greatest common divisor, reserving (a, b) for the ideal generated by a and b. In a PID one does have (a, b) = (gcd(a, b)) so using (a, b) for both the ideal generated by a and b and for their greatest common divisor is perhaps tempting. But for commutative rings with 1 the greatest common divisor does not always exist. In a UFD that is not a PID, the greatest common divisor does exist, but one usually one does not have (a, b) = (gcd(a, b)).