###### 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. ...

$(\Rightarrow)$ 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. ...

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. ...

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 ...

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 ...

This is a simple application of the Problem 220: W...
Question: 1.223

(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 ...

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. ...

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). ...

Let $\phi$ denote the map sending a p...