1
vote
0answers
103 views

Classification of rings between a PID and its field of fractions?

Let $D$ be a PID and let $\mathrm{Frac}(D)$ be its field of fractions. I want to classify the intermediate rings $D\subseteq R\subseteq \mathrm{Frac}(D)$. Theorem: Every such ring $R$ is a ...
1
vote
1answer
134 views

for which truth-operations f can f-membership in a prime ideal be represented by a polynomial?

Nota bene: all rings are supposed commutative with $1$ and all ring homomorphism should be unital. Let $B$ be the set of truth values {True, False}. For a formula $\phi$ denote by $[\phi]\in B$ it's ...
1
vote
1answer
152 views

Is there a prime of height $i$ in support of $H^i_I(R)$?

$I$ is an ideal of a local Noetherian ring $R$ and $i>0$ . Clearly the height of primes in support of $H^i_I(R)$ is at least $i$ The question is if it contains a prime of height $i$, specially ...
1
vote
0answers
662 views

Prime ideals in polynomial rings over integers

Im trying to find a characterization of the prime ideals in the polynomial ring $R = \mathbb Z[X,Y]$ in two variables over the integers. Actually I need to find the maximal ideals in quotient rings ...
2
votes
0answers
265 views

Orthogonality (wrt. Ext, Tor) in commutative noetherian rings

Hi, it is a folklore, that: let $p$, $q$ be two primes of a commutative Gorenstein ring $R$. $$ \operatorname{Tor}^k(E(R/p), E(R/q)) \neq 0 \iff p = q\mbox{ and }k = \operatorname{height} p. $$ ...
0
votes
1answer
180 views

Number of generators of $\mathfrak m$-primary ideals in $k[x, y]$

Let $R = k[x, y]$ with $k$ algebraically closed, and $\mathfrak m = (x, y)$. Suppose $I$ is an $\mathfrak m$-primary ideal of $R$, i.e., $(x, y)^n \subset I \subset (x, y)$ for some $n$. If ...
4
votes
1answer
183 views

Explicitly generating 1 in an ideal without prime support

The Question Let $R$ be a unital commutative ring, and let $a,b_1,b_2\in R$. The following is a basic commutative algebra exercise. Lemma. If $Ra+Rb_1=R$ and $Ra+Rb_2=R$, then $Ra+Rb_1b_2=R$. ...
13
votes
3answers
2k views

What are the prime ideals of k[[x,y]]?

Let $k$ be a field. Then $k[[x,y]]$ is a complete local noetherian regular domain of dimension $2$. What are the prime ideals? I've browsed through the paper "Prime ideals in power series rings" ...