5.2 Principal Ideal Domains
is a subring of the PID
is not a PID (Example 3).
The ideals are 0 =
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.
≠ 0. If a
is a unit then |R
| = 1. Otherwise, by Theorem 4 §3.3, let B
be any ideal of R
, say B
. Then a
. Since a
has a prime factorization, there are at most finitely many such divisors b
up to associates, and hence only finitely ...
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