For questions involving prime ideals in commutative or noncommutative rings.

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### Intersection of powers of prime ideals

Let $R$ be a Noetherian ring. Let $(x)$ be a prime ideal such that $\bigcap_n (x)^n=0$. Then $R$ is a domain.
Is this a known result? I heard its known as the Davis lemma. Can anyone give a ...

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### Does this kind of non-noetherian bimodule exist?

Question: Do there exist simple rings $R$ and $S$ (i.e., rings with no proper nonzero ideals) and an $(R,S)$-bimodule $M$ such that
$M$ is finitely generated both as a left $R$-module and a right
...

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### 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 $R$.
$$
\operatorname{Tor}^k(E(R/p), E(R/q)) \neq 0 \iff p = q\mbox{ and }k = \operatorname{height} p.
$$
...

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134 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 ...

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925 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 ...