5.2 Principal Ideal Domains

1. No. img is a subring of the PID img, but img is not a PID (Example 3).
3. The ideals are 0 =img 0 and F = img 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 = img 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 = img b. Then imgaimgimg 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 ...

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