# Questions tagged [prime-ideals]

For questions involving prime ideals in commutative or noncommutative rings.

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• 923
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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 $R$-...
• 280
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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 ...
• 279
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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 ...
519 views

• 143
1 vote
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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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### Prime elements in integrally closed extension of domains

Let $R \subseteq S$ be an extension of integral domains such that $R$ is integrally closed in $S$. Let $P$ be a prime ideal of $R$. Is $PS$ always a prime ideal of $S$? For classical examples of ...
• 365
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### intersection of two height 2 primes must contain a non-zero prime?

I saw in some contexts the following statement, which I do not have a reference for this: "Kaplansky asked if in a Noetherian domain the intersection of two height 2 primes must contain a non-...
• 1,355
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### Is it true that this ideal must be principal? (proof verification)

Let $L/K$ be a (abelian, Galois) quadratic extension of number fields with $\text{Gal}(L/K)$ generated by $\sigma$ and $\mathfrak{p} = \alpha\mathcal{O}_K$ a principal prime ideal of $\mathcal{O}_K$. ...
1 vote
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• 29
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### 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 ...
• 29
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### Explicit example of prime ideal not an intersection of maximal ideals, in universal enveloping algebra

Let $A$ be a $\mathbb k$-algebra. If $A$ is affine commutative, by Nullstellensatz, then every prime ideal of $A$ is an intersection of maximal ideals. To justify the notion of being primitive in ...
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• 403
1 vote
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• 519
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### On the prime spectrum of $R[[X]]$ when the prime spectrum of $R$ is Noetherian

All rings below are commutative with unity. If $R$ has a.c.c. on radical ideals i.e. if $Spec R$ is Noetherian under Zariski topology, then so is $R[X]$, this is Theorem 2.5 in the following paper ...
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### A relation between $Spec((1+I)^{-1}R)$ and $Spec(R/J)$

Let $R$ be a commutative ring with identity and let $I$ and $J$ be two finitely generated ideals of $R$. Clearly $1+I:=\{1+i:i\in I\}$ is a multiplicative closed subset of $R$. We can consider the ...
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1 vote
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### Localizing prime ideals over Noetherian rings

Let $R$ be a prime Noetherian ring which is not necessarily commutative. Consider the two natural ways to "extend" $R$: $R[x]$ and $M_n(R)$, polynomials over $R$ and $n$ by $n$ matrices over $R$ ...
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1 vote
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### What's the probability a random module element has prime annihilator?

I'm going to pose two versions of my question---an ill-defined version and a well-defined version. Ill-Defined Question (IDQ). Let $R$ be a (commutative, unital) Noetherian ring, $M$ a finitely ...
• 3,059
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### Is the annihilator of a minimal prime ideal principal?

My setup is as follows: $X$ is a projective, reduced curve (which is not integral) with a finite morphism onto $\mathbb{P}_k^1$. $\DeclareMathOperator{\Ann}{Ann}$ Let $R$ be a coordinate ring of $X$ ...
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### Cellular and primary binomial ideals

Let $I \subseteq \mathbb{K}[x_1, \dots, x_n]$ be an ideal of a polynomial ring over a field $\mathbb{K}$. $I$ is called cellular if every variable $x_i$, with $i=1, \dots, n$, is either a ...
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1 vote
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### A special family of prime ideals

I am looking for a commutative ring $R$ with identity which has a family $F:=\{p_i\}_{i\in A}$ of prime ideals such that for each $n\in \mathbb{N}$ there exists $m\in \mathbb{N}$ with $n\leqslant m$ ...
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### For every prime ideal $P$ of any Cohen-Macaulay ring $R$, is the sequence $\operatorname{depth}(R/P^n)$ eventually constant?
Let $P$ be a prime ideal of a Cohen-Macaulay ring $R$. Then is the sequence $\operatorname{depth}(R/P^n)$ eventually constant ?