Math 793
Summer 2001
Homework 1
“Ring”
means commutative with identity unless otherwise specified.
Turn
in at least two of the following problems on Monday, June 18, 2001.
·
divides both
and
·
if
divides both
and
then
divides
a)
Give
an example of a domain and two nonzero
elements of
that have no greatest
common divisor.
b)
Show
that if is a domain where
every ideal is principal (PID or principal ideal domain), then any two nonzero
elements
and
have a greatest
common divisor
and
is an
linear combination of
and
(that is, there exist
such that
).
c)
Show
that if is a PID, then any
nonzero prime ideal of
is maximal.