May 2012
Beginner
160 pages
4h 24m
English
Content preview from Introduction to Abstract Algebra, Solutions Manual, 4th Edition
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5.2 Principal Ideal Domains
1. No. 
is a subring of the PID 
, but 
is not a PID (Example 3).
is a subring of the PID , but is not a PID (Example 3).
3. The ideals are 0 =
0 and F = 1 , both principal.
0 and F = 1 , both principal.
4. No. If it were a PID it would be a UFD by Theorem 1, contrary to Example 5 §5.1.
5. Let A = a , a ≠ 0. If a is a unit then |R/A| = 1. Otherwise, by Theorem 4 §3.3, let B/A be any ideal of R/A, say B = b. Then a
⊆ b so b|a. Since a has a prime factorization, there are at most finitely many such divisors b of a up to associates, and hence only finitely ...
a , a ≠ 0. If a is a unit then |R/A| = 1. Otherwise, by Theorem 4 §3.3, let B/A be any ideal of R/A, say B = b. Then a ⊆ b so b|a. Since a has a prime factorization, there are at most finitely many such divisors b of a up to associates, and hence only finitely ...Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
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