Questions tagged [prime-ideals]
For questions involving prime ideals in commutative or noncommutative rings.
108 questions
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Is this ideal of a polynomial ring over a field a contracted ideal?
Let $k$ be a field and $R:= k[y_1, \dotsc , y_d]$ be a polynomial ring in $d$ variables over $k$. Set $K:= QF(R)$. Given finitely many elements $a_1, \dotsc , a_n$ algebraic over $K$, we consider the ...
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Unknown notation involving prime ideals of number fields
I'm reading the paper
R. R. Laxton, “On a problem of M. Ward,” Fibonacci Quart., 12 pp. 41–44 (1974),
which can be downloaded for free from the Fibonacci Quarterly website: https://www.fq.math.ca/12-...
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Choosing points in an affine space in general position
Let $k$ be a field and $A$ a finitely generated commutative $k$-algebra. Let $p \subseteq A$ be a non-maximal prime ideal and $m_i \subseteq A, i \in I$ a collection of maximal ideals such that $m_i \...
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Primes avoiding a specific ideal in the preimage
Let $f \colon A \to B$ be a map between to regular local rings such that $\dim(A)=\dim(B)+1 \geq 3$, let $\mathfrak{p} \subset B$ be a non-maximal prime ideal, and let $\mathfrak{a} \subset A$ be an ...
- 1,103
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Sieving in the Gaussian integers
I asked this question as below a couple weeks ago in stackexchange but got no comments/answers, so I'll ask it here. (My understanding was that this is ok? Let me know if not).
Question: Can anyone ...
- 1,808
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Action of torus on Laurent polynomials
Let $F$ be an algebraically closed field and suppose that the torus $(F^*)^n$ acts on the Laurent polynomial ring $L$ in $n$ variables $X_1, \dots, X_n$ defined by $X_i \dashrightarrow a_iX_i$ for ...
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Finiteness of Krull dimension of commutative Noetherian ring for which maximal length of regular sequence in maximal ideals have a uniform upper bound
$\DeclareMathOperator\grade{grade}$Let $R$ be a commutative Noetherian ring. For an ideal $I$ of $R$, let $\grade(I,R)$ be the maximal length of an $R$-regular sequence in $I$.
My question is: If $\...
- 413
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Real exponentiation in the quotients of rings of continuous functions by prime ideals
Consider the ring $C = C(X) = C(X; \mathbb{R})$ of continuous functions $f:X\to \mathbb{R}$ where $X$ is a Tychonoff space. This is naturally a lattice ordered ring by setting $f\geq 0$ iff $f(x)\geq ...
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$S/I$-freeness of $I/I^2$ vs $I/I^{(2)}$, where $I$ is a radical ideal of regular local ring $S$
Let $I$ be a radical ideal of a regular local ring $S$. Put $R:=S/I$. Let $I^{(n)}$ be the $n$-th symbolic power of $I$. It is well-known that $I^n \subseteq I^{(n)}$.
Is it true that $I/I^2$ is $R$-...
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Irreducibility of an explicit complex projective variety
Let $Y\subset \mathbb P^n_\mathbb C$ be a subvariety defined by a series of homogeneous polynomials $f_1, \ldots, f_t$. Is there an effective way to determine the irreducibility of $Y$ as an algebraic ...
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Generalizations for the PNT to a subset of Dedekind domains?
The classical prime number theorem states that the prime counting function
$$\pi(X) := \# \{ p \leq X \ | \ \text{$p$ prime} \}$$
is asymptotically equal to $X/\log(X)$.
It is also known (and much ...
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Density of prime ideals of a given degree
Let $K$ be a number field. For each ideal $I$ of the ring of integers $\mathcal{O}_K$ let $N_K(I)$ denote the norm of $I$. For a prime $\mathfrak{p}\subset \mathcal{O}_K$ above the rational prime $p\...
- 605
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Closed prime ideal in $C[0, 1]$
I know that maximal ideals of $C[0, 1]$ corresponds to singleton. Also, using Zorn's lemma one can construct a prime ideal in $C[0, 1]$ which is not maximal.
Is there any $\textbf{closed}$ prime ...
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Finite type injective ring map between domains preserves the open point $(0)$
I am looking for a proof of the following statement without using full power of Chevalley's theorem on constructible sets. We say a domain $A$ is $0$-open if $\{(0)\}$ is open in $\operatorname{Spec}(...
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Ideals whose alebraic variety is a singleton
I do not work in algebra, so i apologize in advance if there are some unclear/wrong sentences. Let us consider the ring $\mathbb{C}[X_1,\ldots,X_q]$ of polynomials in $q$ variables. For an ideal $I$ ...
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