Abstract Algebra notes
maximal ideal:
(253)
- ideal M in S is maximal if M!=S and only ideals containing M
are M and S
- every maximal ideal is prime
- ideal M is maximal iff R/M is field
-
prime ideal:
- every nonzero prime ideal in a P.I.D. is maximal
division ring:
(224)
- ring R is division ring if every nonzero element has a
multiplicative inverse
(every nonzero element is a unit)
field:
(224)
- commutative division ring
- fields contain no zero divisors
- Z/nZ is field iff n is prime
- any finite integral domain is a field (228)
- fields are euclidean domains (271)
euclidean domain:
- a euclidean domain is a P.I.D.
Euclidean Domains
\definition{Euclidean Domain} Integral domain is
if there is a norm
on
such that for any two elements
and
of
with
there exist elements
and
in
with
Every ideal in a Euclidean Domain is principal. If is any nonzero ideal in the Euclidean Domain
then
, where
is any nonzero element of
of minimum norm.
If is the ideal of
generated by
and
, then
is a greatest common divisor of
and
if
(i) is contained in the principal ideal
, and
(ii) if is any principal ideal containing
then
.
If two elements and
of
genearte the same ideal, then
for some unit
.