All Questions
Tagged with ac.commutative-algebra prime-ideals
8 questions
17
votes
4
answers
2k
views
Constructive proof that a kernel consists of nilpotent elements
I am interested in the following innocent looking statement:
Let $A \leftarrow R \rightarrow B$ be two homomorphisms of commutative rings. Assume that their kernels consist of nilpotent elements. ...
- 5,482
9
votes
2
answers
364
views
When $C (X) $ is zero dimensional
Let $X $ be a Tychonoff topological (completely rgular) space and $C (X) $ be the ring of all real valued functions over $X $. When is the krull dimension of $C (X) $ zero?
- 105
8
votes
1
answer
1k
views
When does prime elements remain prime in certain integral extension
Let $R$ be an integral domain and $\bar R$ denote its integral closure in the fraction field (i.e. normalization). If $p\in R$ is a prime element in $R$, then does $p$ remain prime in $\bar R$ also ?
...
user111492
4
votes
1
answer
364
views
Values attained by the coheight of $(H \setminus H^\times)^k$ as a function of $H$ and $k$
Edit (Apr 24, 2017). I'm updating this post in the light of the latest developments of a related thread.
Let $H$ be a multiplicatively written, commutative monoid, and set $M := H \setminus H^\times$,...
- 10.5k
3
votes
2
answers
165
views
Weak ideal systems $r$ for which the $r$-coheight satisfies a kind of triangle inequality
Let $H$ be a multiplicatively written, commutative monoid with identity $1_H$, and let $\mathcal P(H)$ be the power set of $H$. If $X, Y \subseteq H$, we will set $$XY := \{xy: x \in X,\, y \in Y\}.$$
...
- 10.5k
9
votes
1
answer
443
views
Rings with all non-prime ideals finitely generated
Motivated by this question, I would like to ask:
If all non-prime ideals in a ring are finitely generated, then is the ring Noetherian? Can we at least say anything in the local case?
Note that ...
user111492
4
votes
0
answers
908
views
Methods to check if an ideal of a polynomial ring is prime
Fix $\ell \geq 3$, $r \geq 2$ and $1 \leq k \leq \ell - 1$ and $z_1, \ldots, z_\ell \in \mathbb{C}$ with $z_i \neq 0$ for all $i$ and $z_i \neq z_j$ for all $i \neq j$. Now consider the (irreducible, ...
- 61
2
votes
0
answers
524
views
Irreducibility over the field of fractions of a quotient of a polynomial ring
Fix $\ell \geq 3$, $r \geq 2$ and $1 \leq k \leq \ell - 1$ and $z_0, \ldots, z_\ell \in \mathbb{C}$ with $z_i \neq 0$ for all $i$ and $z_i \neq z_j$ for all $i \neq j$. Now consider the (irreducible, ...
- 61