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 ...
Get Introduction to Abstract Algebra, Solutions Manual, 4th Edition now with the O’Reilly learning platform.
O’Reilly members experience live online training, plus books, videos, and digital content from nearly 200 publishers.