Math
720
Fall 2000
Exam 2
The
purpose of these exercises is to show that the dimension of a polynomial ring
does not get “much bigger” than the dimension of its coefficient ring. In this
exam you may assume that all rings are commutative with identity and S is a
multiplicatively closed set not containing 0. The result that we will obtain is
that if R is a commutative ring with 1 such that dim(R)=n, then the dimension
of R[x] is between n+1 and 2n+1. Remember to start counting at 0 where Krull
dimension is concerned.
1.
Let 
be an ideal and let 
be a ring containing 
We define the extension of I to T to be the ideal 
(that is, the ideal
generated by 
in 
a)
Show that 
Is the analogous
result true for power series?
b)
Show that if 
is a prime ideal,
then the ideals 
(respectively 
) are prime in 
(respectively 
).
c)
Show that the following are isomorphic:
·
·
2.
Let 
be a multiplicatively closed subset of 
Show that 
Is the analogous
result true for power series?
We now assume that there are (at least) three prime
ideals of R[x] in a chain lying over a fixed prime ideal of R. That is, we have
such that 
for i=0,1,2.
3.
(Reduction
of problem I) Consider the situation above:
a)
Use 1a) to show that 
must contain 
b)
Continue to use the above results to show that if 
is lain over by at
least three primes in a chain from 
then you can assume
that 
c)
Use parts a) and b) and problem 1 to show that you can
reduce the problem to the case where 
is an integral domain. More specifically, we can reduce to
the statement below.
We have reduced to the case where R is an integral
domain. In this case we have (at least) three prime ideals of R[x] in a chain 
such that 
for i=1,2.
4.
(Reduction
of problem II) In our new situation prove the following:
a)
Show that if 
is the set of nonzero elements of 
then 
is a principal ideal
domain.
b)
Show the set 
has empty intersection with the ideals 
c)
Show that in 
there must be a chain
consisting of at least three prime ideals. Obtain a contradiction.
Finishing touches.
5.
Given a commutative ring with identity such that 
show the following:
a)
Given a chain of primes of length 
in 
exhibit a chain of length 
in 
b)
In problems 3 and 4 we showed that there cannot be three
primes of 
in a chain lying over any fixed prime in 
Use this information
to show that 