For questions involving prime ideals in commutative or noncommutative rings.

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8
votes
0answers
198 views

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 ...
2
votes
0answers
290 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 $R$. $$ \operatorname{Tor}^k(E(R/p), E(R/q)) \neq 0 \iff p = q\mbox{ and }k = \operatorname{height} p. $$ ...
1
vote
0answers
136 views

A family of maximal ideals

Let $m_i $, $i \in I,$ be an infinite family of maximal ideals in a commutative ring with identity (it is not supposed to be Noetherian). When does there exist $j \in I$ such that $\cap_{i\not= j} ...
1
vote
0answers
177 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
0answers
1k 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 ...
0
votes
0answers
96 views

Can it occur that $q^{ce}$ is a prime ideal (of $S$), while $q^{ce}\neq q $?

Let $R$ and $S$ be commutative rings (with $1$) and $f : R\to S$ be a ring homomorphism. For an ideal $I$ of $R$, set $I^e:=\langle f(I)S\rangle$ (called the extension of $I$ to $S$) and for an ideal ...