Questions tagged [prime-ideals]

For questions involving prime ideals in commutative or noncommutative rings.

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17 votes
4 answers
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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. ...
HeinrichD's user avatar
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9 votes
2 answers
350 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?
Azitro Walex's user avatar
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 ? ...
user avatar
4 votes
1 answer
348 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 $M := H \setminus H^\times$,...
Salvo Tringali's user avatar
3 votes
2 answers
160 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 $$XY := \{xy: x \in X,\, y \in Y\}.$$ ...
Salvo Tringali's user avatar
9 votes
1 answer
428 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 ...
user avatar
4 votes
0 answers
818 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, ...
Nils Amend's user avatar
2 votes
0 answers
494 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, ...
Nils Amend's user avatar
1 vote
0 answers
181 views

Do $k$ specific prime factors uniquely determine the continuous composite sequence of length $k$?

猜想:不存在两个长度为k且一共含有k个不同素因子的连续合数序列的素因子集合相等。例如 24 25 26 27 (2 3 5 13) 其余长度为4的连续合数序列要么素因子个数大于4 要么素因子集合不等于{2 3 5 13} 这个连续合数猜想的重大特定情况下的猜想。难度可与哥德巴赫猜想匹敌,非天才误入。再比如2 3 5 唯一决定了8 9 10 除此之外再无其他。总之经过计算机验证没有找到反例。 ...
光子精灵S's user avatar
0 votes
0 answers
50 views

k specific prime factors guess and related prime guess [duplicate]

there is no more than one group of continuous composite sequence of length k composed of only k different specific prime factors. for example 2 3 5[8 9 10]just only one group. I have prove that k ...
光子精灵S's user avatar