Questions about commutative rings, their ideals, and their modules.

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Intersection of modules is equal to product.

If $B$ is a commutative ring and let $\mathcal{Q}_1,\ldots,\mathcal{Q}_n$ ideals relative primes. Let $M$ be a $R$-module. I don't sure if this is true. Then $$(\mathcal{Q}_1\cap\cdots ...
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26 views

Describe the differential of $d\phi : \mathbb{T}_{t, \mathbb{A^1}} \rightarrow \mathbb{T}_{t^3, t^4, t^5, W}$

Let $V = \mathbb{A^1}$ and $W = Z(xz - y^2, yz, x^3, z^2 - x^2y) \subset \mathbb{A^3}$ and let $\phi: V \rightarrow W$ be a surjective morphism, describe the differential. Currently in a course in ...
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1answer
117 views

Finding a finite generating set of an ideal of monomials

My problem involves considering the ideal $I = \{ X^mY^n \mid m,n\in \mathbb{N}, m^2n>5 \}$ of $\mathbb{Q}[X, Y]$. I am asked to write down a finite generating set of $I$ and explain how I ...
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1answer
35 views

Want to show that $g\in I$ where $I$ is an ideal, given the following conditions

Let $R=K[x_1,...,x_n]$ and $I$ be an ideal of $R$, $K$ being a field Given $h\in I$, $g\in \sqrt{I}$ and $f\in\sqrt{I}$ Where $in_<(f)=in_<(h)$ and $g=f-h$. So $in_<(g) < ...
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2answers
30 views

Relation between Variety of $(I\cap J)$ and Variety of $(I)$ $\cap$ Variety of $(J)$

I was wondering whether a relationship exists between $V(I\cap J)$ and $V(I)\cap V(J)$. Where $I$ and $J$ are ideals of the ring $R=K[x_1,...,x_n]$.
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1answer
80 views

Is the intersection of two Noetherian rings Noetherian?

Is the intersection of two Noetherian rings also Noetherian? If yes, could you please give me the idea of proof. If not, give me an counterexample.
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1answer
26 views

$\mathbb{Q}[t]$ is integrally closed in $Quot(\mathbb{Q}[t])$

I'm having trouble trying to show that $\mathbb{Q}[t]$ is integrally closed in $Quot(\mathbb{Q}[t])$. Where $Quot(\mathbb{Q}[t])$ is the field of fractions of $\mathbb{Q}[t]$. So I'm trying to show ...
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1answer
31 views

Line Bundles on Local Rings

Let $A$ be a local ring and $L$ a module over $A$ which is projective and of rank one. Does it follow that $L$ is isomorphic to $A$?
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1answer
37 views

Finding $\mathbb{Z}[\sqrt{-3}]/(p)$ for some prime $p$.

I have to prove that $\mathbb{Z}[\sqrt{-3}]/(p)\cong \mathbb{F}_{p^{2}}$ if $p\equiv 5\ \text{mod}\ 6$ and $\mathbb{Z}[\sqrt{-3}]/(p)\cong \mathbb{F}_{p}\oplus\mathbb{F}_{p}$ if $p\equiv 1\ ...
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1answer
24 views

Showing that $in_<(f^m) = in_<(f)^m$

I am currently in the following scenario: Let $I$ be an ideal of $K[x_1, ..., x_n]$, $<$ be a fixed term order and $in_<(I)$ be radical. I want to show that: $in_<(f^m) = in_<(f)^m$ ...
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1answer
35 views

Proof of Unique factorization in Dedekind Rings .

Proof de unique factorizaation in Dedekind Rings. Algebraic Number fields, Janusz, Second edition. In the above proof, Theorem 3.13. Why of the corolary 3.7, ...
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1answer
68 views

“Going between” property

Let $A \subset B$ be an integral ring extension and assume that $A$ is a finitely generated $K$-algebra over some field $K$. Let $P_1\subsetneq P_3$ be prime ideals of $A$ and let $Q_1\subsetneq ...
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1answer
57 views

Counterexample to the finiteness of integral closure of a Dedekind domain.

Let A be a Dedekind domain, K its field of fractions, L/K a finite extension, B the integral closure of A in L. By the Krull-Akizuki theorem, B is noetherian, hence B is a Dedekind domain. In the ...
2
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2answers
113 views

$X$ compact Hausdorff space, characterize the maximal ideals of $C(X)$

I know this question has been asked before, but I think I'm very close to a new solution and wanted to know if it is a viable approach. Let $C(X)$ be the ring of continuous functions $X \rightarrow ...
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3answers
177 views

``Minimal generating ring" for a field of fractions

In this answer and the linked MathOverflow post, it's shown that any field $F$ of characteristic zero contains a proper subring $A$ such that $F$ is the field of fractions of $A$. However, there is ...
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1answer
38 views

Viewing the universal property of rings of fractions as a universal arrow

For a multiplicatively closed subset $S$ of $A$, we have a functor $S^{-1}: A-Mod \rightarrow S^{-1}A-Mod$. I am trying to understand this functor a little bit better and I was thinking about the ...
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1answer
43 views

How to see $\Gamma(\mathscr{O}_S,\operatorname{Proj} S)$ as a ring?

Let $S$ be a finitely generated graded $A$-algebra. For each homogeneous $f\in S_+$, we have a scheme structure $D(f)\cong \operatorname{Spec} S_{(f)}$ where $S_{(f)}$ denotes the zeroth piece of the ...
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1answer
30 views

$A\subset B $ with $B$ integral domain. If $B$ is integral over $A$ can we say that $Q(B)$ is algebraic over $Q(A)$?

Let $A\subset B$ with $B$ an integral domain. If $B$ is integral over $A$ can we say that $Q(B)$ is algebraic over $Q(A)$ ? (Here $Q(\dots)$ denotes the quotient field of $(\dots))$.)
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Is $\mathbb{Z}$ the only totally-ordered PID that is “special”?

(All my rings are commutative and unital.) Definition. Call a totally-ordered ring $R$ special iff for all non-zero $b \in R,$ every coset of $bR$ has a unique element in the interval $[0,|b|).$ ...
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1answer
84 views

Prove that $R[\sqrt{\pi}]$ is a DVR

If $R$ be a DVR(discrete valuation ring) with uniformizer $\pi$, then prove that $R[\sqrt{\pi}]$ is a DVR. How shall I begin, first do I have to find a candidate for the uniformizing element of ...
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1answer
40 views

A closure of associated points as a support of an element of $m \in M$ where $M$ is a finitely generated $A$ module

Let $A$ be a Noetherian ring and $M$ a finitely generated $A$ module. Suppose $p_1, p_2, p_3 \in Ass (M)$, some associated primes of $M$ such that $p_i = ann (m_i)$. I wanted to show that there ...
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0answers
107 views

What about $\mathrm{Spec}(\mathbf{Q})$?

I've heard a lot about $\mathrm{Spec}(\mathbf{Q})$ (see for example Minhyong Kim's answer here), but $\mathbf{Q}$ is a field. So isn't $\mathrm{Spec}(\mathbf{Q})$ trivial? What's the point of studying ...
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1answer
61 views

Elements of a localization

How does a localization at a prime look like, for example if we have $R:=\mathbb Z[\sqrt{-3}]$ and let the ideal $\mathfrak p:=\left(\sqrt{-3}\right)$ in $R$, what are the elements of $R_{\mathfrak ...
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1answer
121 views

Showing that if the initial ideal of I is radical, then I is radical.

I need to show that, given a term order $<$ and an ideal $I$, if $in_<(I)$ is radical, then $I$ is radical. Any help or hints would be appreciated as I'm not really sure where to start, ...
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0answers
42 views

Inclusion of fractional ideals implies equality

Let $R$ be a integral domain and let $\mathfrak U\subseteq\mathfrak B$ two ideals of $R$ such that $\mathfrak UR_\mathfrak p=\mathfrak BR_\mathfrak p$ for all maximal ideals. Then $\mathfrak ...
2
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0answers
38 views

indecomposable summand

Let $M$ be an $R$ module. Is this true for start a proof that we say "Let $S$ denote the indecomposable summands of $M$"? In fact, I want to know whether any module over a Dedekind domain (or a ...
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1answer
52 views

Vanishing set of $\text{Ann} (M)$, where $M$ is a finitely generated $A$ module

Let $M$ be a finitely generated $A$ module, generated by say $x_1, ..., x_n$. Let $V(S)$ denote the set of primes of $A$ containing $S$. I am guessing that $$ V(\text{Ann}(M)) = \cup_{1 \leq i \leq ...
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2answers
38 views

Isomorphism of the completion of polynomial ring modulo second degree polynomial

Let $k$ be a field of characteristic different from $2$, and $A=k[x,y]/(y^2-x^2(x+1))$. Let $\hat A$ be the $(x,y)A$-adic completion. How can I show that $\hat A\simeq k[[u,v]]/(uv)$? Qing Liu: ...
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0answers
35 views

Is submodule of Hilbert module a Hilbert module?

Let $R$ be a commutative ring with identity and $M$ be an $R$-module. $M$ is a Hilbert module if every prime submodule $P$ of $M$ equals the intersection of all maximal submodules of $M$ that contain ...
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1answer
53 views

Going Up Theorem and Affine Sets.

So for an affine scheme, we know that this is true: Suppose that $k$ was algebraically closed. Let $X$ and $Y$ be affine schemes and $\phi: X \rightarrow Y$ be a polynomial map with the corresponding ...
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2answers
105 views

Computing the Grothendieck group of affine space.

For a Noetherian scheme $X$ the Grothendieck group $K(X)$ is defined as the free abelian group on coherent sheaves quotiented by the equivalence relation $\mathscr{F}=\mathscr{F}'+\mathscr{F}''$ for ...
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0answers
42 views

Homological criterion for $A(B \cap C) = AB \cap AC$?

Is there a homological criterion for the condition $A(B \cap C) = AB \cap AC$ for ideals in a ring $R$? By "homological" I mean a statement such as "the given equation holds if and only if (some Tor, ...
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1answer
40 views

Ring contained in a R-module finitely generated

Let $R$ be a Noetherian domain with quotient field $K$ and let $b_1,\ldots,b_n\in K$. Suppose that $R'$ is a integral domain, $R\subseteq R'$ and $$R'\subseteq \sum_j Rb_j.$$ Remark: It is ...
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0answers
48 views

Extension of an ideal to a subring of the ring of fractions

Let $A$ be a domain, and $B$ an $A$-algebra inside $\text{Frac}(A)$. Let $x/y\in B$. Then $(yA:_Ax)B\neq B$ if and only if there is a prime ideal $\mathfrak{p}\in \text{Spec}(A)$ such that ...
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1answer
105 views

Integral closure and field of fractions

I have a ring $R = \mathbb{Q}[t^2,t^5] \cong \frac{\mathbb{Q}[x,y]}{\langle x^5 - y^2 \rangle}$ (where the denominator is the ideal generated by $x^5 - y^2$). Now i have to compute the closure of $R$ ...
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2answers
98 views

Relation between intersection and product of ideals

Let $C$ be a smooth projective (irreducible) curve in $\mathbb{P}^n$ for some $n$. Denote by $I_C$ the ideal of $C$. Let $g \in I_C\backslash I_{C}^2$, an irreducible element. Is it true that for any ...
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1answer
30 views

Reference for Hilbert function and multiplicities

What is the good book/notes for Hilbert function and multiplicities( except BRUNS and MATSUMURA )? I need basic as well as advance level
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0answers
58 views

Classical algebraic geometry in infinite dimensions?

I ran into this paper of Serge Lang, if I understood him correctly (of which I am doubtful), then since $\mathbb{C}$ has uncountable transcendence degree over its base field $\mathbb{Q}$, Hilbert's ...
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1answer
45 views

Finding the kernel of a multiplication map

Consider the ideal $I=(x,y) \subset R=\mathbb{C}[x,y]$ and $\mathbb{C}$ as the $R$-module $R/I$. I am asked to find the kernel of the multiplication map $I \otimes_R I \rightarrow I$ as a submodule of ...
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1answer
64 views

Overring of an integrally closed domain that is not integrally closed

Assume that $A$ is an integrally closed integral domain, and $K$ is its fraction field. Well...this may be a stupid question, but is every overring of $A$ between $A$ and $K$ also integrally closed ? ...
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41 views

Commutative ring where $r$, $s$ are associates but $r \neq us$ for any $u$ unit. [duplicate]

First of all I think it's important to note that the definition of associates that $r$ divides $s$ and $s$ divides $r$. Secondly, I know that my ring $R$ has to have zero divisors since if $R$ is an ...
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1answer
40 views

Localization of Tor

I have few questions about the second part. (1) I'm not sure why $\operatorname{Tor}_1^A(M,\bar{A}) \otimes_B B_P=\operatorname{Tor}_1^A(M_P,\bar{A})$. (2) I think $\bar{A}$ has a free ...
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1answer
78 views

Theorem 12.3 from Matsumura

Theorem 12.3 (p. 87), Commutative Ring Theory by Matsumura. Let $A$ be a Krull ring, $K$ its field of fractions, and $\mathfrak{p}$ a height $1$ prime ideal of $A$; then if $\mathcal{F} = ...
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1answer
74 views

$\varinjlim\operatorname{Hom}_R(N,M_i) = \operatorname{Hom}_R(N, \varinjlim M_i)$

Show that $\varinjlim \operatorname{Hom}_R(N,M_i) = \operatorname{Hom}_R(N, \varinjlim M_i)$ is true when $N$ is finitely generated and $R$ is noetherian. Do you think the noetherian condition is ...
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1answer
31 views

The functor Tor for $r_R$

Suppose $R$ is commutative ring and $r \in R$. Show that if $r$ is a zero divisor, then $$\text{Tor}^R_n(R/(r),M) \cong \text{Tor}^R_{n-2}(r_R,M)$$ for $n\geq 3$, where $r_R =\{s \in R \ |\ rs =0 \}$. ...
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0answers
32 views

If P(X) is reducible in K[X], show it is reducible in A[X], A integrally closed domain

Let $A$ be an integrally closed ring, $K$ its field of fractions, and $P(X) \in A[X]$ a monic polynomial. If $P(X)$ is reducible in $K[X]$, show that it is reducible in $A[X]$. The hint given is to ...
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1answer
71 views

Ideal Quotient and Zero Locus.

I stumbled across something I couldn't get while reading. So given two ideals $I$ and $J$ in some ring $R$ where $R = k[\mathbb{A}^n]$. I want to show that $Z(I) - Z(J)$ $\subset$ $Z((I:J))$, where ...
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0answers
49 views

Image of ideal under the isomorphism given by the Chinese Remainder Theorem.

Suppose that $\mathfrak{p}_1,\ldots,\mathfrak{p}_n$ are maximal ideals of a ring $R$. Then $\mathfrak{p}_i+\mathfrak{p}_j=R$ with $i\neq j$ and $\mathfrak{p}_i^a+\mathfrak{p}_j^b=R$ with $a,b$ ...
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0answers
28 views

Is the ring $F(U(R))$ necessarily isomorphic to the ring of all polynomials with coefficients in $R$ and with constant term equal to $0$?

(All my rings are commutative, but not necessarily unital.) I was playing around with the ring freely generated by an Abelian group, and it seems to me that the following holds: letting $U$ denote ...
0
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1answer
29 views

How to compute singular points on a variety?

Let $H$ be the variety defined by $H = \{(x, y, z, t)\in \mathbb{C}^4: xy = z^2 + t^2\}$. How to compute all singular points on $H$? Thank you very much. My partial solution: it seems that ...