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.
- Let
be a ring and 
a multiplicatively
closed subset of 
(with 
). 
is said to be saturated
if it contains all divisors of elements in 
(i.e. 
). Show that 
is
multiplicatively closed and saturated if and only if the set
complement of
in 
is a union of
prime ideals. - Give an example of a
ring that is not a field that contains no (nonzero) irreducible elements.
- Given
(not both 0) we
define the greatest common divisor of a and b (gcd(
)) to be an element 
satisfying:
·
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.