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

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Prime and Maximal Ideals

I have proved that $<x>$ is a prime but not maximal ideal in $\mathbb{Z}$[x]. I am asked to prove I is maximal in $\mathbb{Z}$[x]. $\\$ I = {$f$ $\in$ $\mathbb{Z}$[x] : the constant term of $f$ ...
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1answer
26 views

An ideal with homogeneous radical is homogeneous

Let $I$ be an ideal of a graded ring $A$. Is it possible that $rad(I)$ is an homogeneous ideal of $A$, but $I$ is not homogeneous?
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22 views

Comultiplication in graded Hopf Algebras

Let $H$ be a graded Hopf algebra over some commutative ring $k$. I'm looking for a proof of the following result, which seems to be stated in various locations. For $x$ in $H$ of degree $n$ ...
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2answers
31 views

Why is not the polynomial ring $R[x]$ a unique factorization domain, where $R$ is the quadratic integer ring $\mathbb{Z}[2\sqrt{2}]$?

Why is not the polynomial ring $R[x]$ a unique factorization domain, where $R$ is the quadratic integer ring $\mathbb{Z}[2\sqrt{2}]$? I'm trying to find a irreducible nonprime element or something but ...
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1answer
22 views

Finding a specific module.

I want to think of a module $M$ over a commutative ring with identity $R$ such that $M \oplus N = R^3 $ while $N$ is isomorphic to $M$. Are there some interesting examples which satisfies this ...
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22 views

Finitely generated torsion free $R$-modules [on hold]

Every finitely generated torsion free $R$-module embed into a finitely generated free $R$-module when total quotient ring of $R$ is a direct product of fields.
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1answer
16 views

An $R$-algebra $S$ is finite over $R$ iff $S$ is generated as an $R$-algebra by finitely many integral elements.

This is a proof from Eisenbud's Commutative Algebra with a View towards Algebraic Geometry. I don't understand the proof of the converse statement, namely the place where Proposition 4.1 is used. ...
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2answers
34 views

A question on a problem on localization from Atiyah (3.8)

I was having trouble with the following problem from Chapter $3$ of Atiyah-MacDonald Let $S, T$ be multiplicatively closed sets in the ring $A$, such that $S\subseteq T$. Let $\varphi : S^{−1}A \to ...
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2answers
37 views

Why rings of fraction is defined as that?

Let $R$ be the commutative ring, S be the multiplicative set. As we know in $S^{-1}R$, $a/b\sim c/d$ iff there is a $t \in S$ such that $adt=bct$. The question is that why we need $t$? Why not just ...
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1answer
32 views

Basic open sets in the Zariski topology are also compact.

Let $A$ be a commutative ring and $X = \text{Spec}(A)$. The closed sets are those of the form $V(E) = \{$ prime ideals $\hat{p} \subset A $ containing $E \}$. And the open sets are the complements ...
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1answer
13 views

Extending and contracting an ideal by a faithfully flat homomorphism

Let $ B $ be a faithfully flat $ A $-algebra. Let $ I \subset A $ an ideal. Shows that $ IB \cap A = I $. This is the second item of Exercise 2.6, Chapter 1, of the Qing Liu's book Algebraic Geometry ...
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1answer
36 views

Suppose that $R$ is a one-dimensional normal Noetherian local ring. Then the maximal ideal $m_R$ is principal

Theorem 11.2 (Matsumura's Commutative Ring theory) gives us equivalent conditions for a ring $R$ to be considered a DVR. I was stuck while reading the proof of $(4) \implies (3)$, (3) $R$ is a ...
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0answers
29 views

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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0answers
22 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
36 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
30 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
25 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
56 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
19 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
24 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
30 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
22 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
30 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
56 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
36 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 ...
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2answers
91 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
166 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
34 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
37 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
28 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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1answer
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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 positive $b \in R$ and any coset $C$ of $bR$, we have that $C$ has a unique element in ...
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1answer
77 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
37 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
85 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
54 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
108 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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39 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 ...
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1answer
50 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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Isomorphism of the completition 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
33 views

Invertible element of a $p$-adic integers [closed]

Let $A=\mathbb Z_p$, $I$ an ideal of $A$ such that $A$ is a complete ring for $I$-adic topology. What would be an example of $n\geq 2$ such that $n$ is invertible in $A$?
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Example of a complete ring [closed]

What would be an example of a commutative ring $A$ with unit and its ideal $I$ such that $A$ is a complete ring with $I$-adic topology?
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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
46 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
89 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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40 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
38 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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1answer
38 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
97 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
93 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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0answers
28 views

Embedding of torsion free module into free module [closed]

Assume $R$ is a Noetherian regular local ring. Can a finitely generated torsion free $R$-module $M$ be embedded in a finitely generated free $R$-module $F$?