Problems and Solutions in Commutative Algebra 1st. Edition by Mahir Bilen Can

Problems and Solutions in Commutative Algebra

227 SOLVED PROBLEMS

Question: 1.227

Prove that if R is a DVR, then R is a PID. Conversely, if R is a PID with a unique maximal ideal, then R is a DVR. ...

Verified Answer:

(\Rightarrow)

Let

K

...

Question: 1.226

Suppose R is a Noetherian integral domain. We assume further that R is a local ring of Krull dimension 1. Prove that if M is the unique maximal ideal of R and if every ideal is a power of M, then R is a DVR. ...

Verified Answer:

Let

x_{1}, \ldots, x_{n} \in M

be a...

Question: 1.225

Suppose R is a Noetherian integral domain. We assume further that R is a local ring of Krull dimension 1. Prove that if M is the unique maximal ideal of R and dimR/M M/M² = 1, then R is a DVR. ...

Verified Answer:

Since

R / M

is a field and since [l...

Question: 1.224

Suppose R is a Noetherian local ring. Prove that if M is the unique maximal ideal of R and dimR/M M/M² is a finite dimensional vector space over R/M. If d = dimR/M M/M², then prove that any generating set for M has at least d element ...

Verified Answer:

Let

U

denote

M / M^{2}[/late...

Question: 1.222

Let S be an integral extension of R. Suppose P1 ⊆ · · · ⊆ Pn is a sequence of prime ideals of R and suppose that there exists a sequence of prime ideals Q1 ⊆ · · · ⊆ Qm of S such that Qi ∩ R = Pi for i = 1, . . . , m. Here, 1 ≤ m < n. Prove that there exists prime ideals Qm+1, . . . , Qn in S such ...

Verified Answer:

This is a simple application of the Problem 220: W...

Question: 1.223

Let k be a positive integer. Prove that the following equalities are true for two ideals I and J from a ring R: (a) I^k ⊆ J ⇒ rad(I) ⊆ rad(J); (b) I^k ⊆ J ⊆ I ⇒ rad(J) = rad(I); (c) rad(IJ) = rad(I ∩ J) = rad(I) ∩ rad(J); (d) rad(rad(I)) = rad(I); ...

Verified Answer:

(a) We start with an easy observation: if

I...

Question: 1.221

Suppose P is a maximal ideal of R, and S is ring that contains R. If there exists a finite number of elements s1, . . . , sn ∈ S that generate S as a ring over R (that is to say, the coefficients of the polynomial expressions in s1, . . . , sn are all from R), then prove that there are only ...

Verified Answer:

By Problem 220 we know that

S

has a...

Question: 1.220

Let R ⊂ S be a pair of rings such that S is integral over R. If P is a prime ideal in R, then prove that there exists a prime ideal Q in S such that P = Q ∩ R. ...

Verified Answer:

The idea is to analyze the situation locally. Let ...

Question: 1.219

Let P be a prime ideal in R and let D = R − P denote the corresponding multiplicative submonoid. Prove that there exists a unique maximal ideal in D^−1R (that is generated by P in D^−1R). ...

Verified Answer:

We know from Problem 218 that prime ideals of [lat...

Question: 1.218

Let D be a multiplicative submonoid of R. Prove that there exists one-toone correspondence between prime ideals not intersecting D in R and prime ideals of D^−1R. ...

Verified Answer:

Let

\phi

denote the map sending a p...