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Questions tagged [commutative-algebra]

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

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Is extending an ideal through a ring homomorphism the same as through extension of scalars?

Suppose we have commutative rings $A$ and $B$, a (maybe injective) ring homomorphism $f: A \rightarrow B$ and and ideal $I \subseteq A$. Is it true that $I^e \cong I \otimes_A B$, where $I^e$ denotes ...
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9 views

On valuation rings satisfying descending chain condition on radical ideals

Let $R$ be a Valuation ring (https://en.wikipedia.org/wiki/Valuation_ring) satisfying d.c.c. on radical ideals. Then , is $R$ a Noetherian ring ? Or at least, does $R$ satisfy a.c.c. on radical ideals ...
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26 views

isomorphism between an ideal and its double dual

Let $R=\mathbb{Z}[\sqrt{-14}]$, $I_1 = (3, 1 + \sqrt{-14})$ and $I_2 = (3, 1 - \sqrt{-14})$. I need to check that the natural maps $I_1 \longrightarrow I_1^{\vee\vee}$ and $I_2 \longrightarrow I_2^{\...
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0answers
14 views

Ring extension where primes can be lifted is integral?

Let $B/A$ be a ring extension of unital commutative rings. Suppose for each prime $p\subset A$, there is $q \subset B$ prime with $q \cap A = p$. It is not true that $B$ is integral over $A$, for ...
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1answer
26 views

Let $f$ be a real irreducible hypersurface of $R^n$. Is $I(V(f))=(f)\subset R[x_1,\dots, x_n]$?

Let $V(f)$ be a real irreducible codimension 1 hypersurface of $R^n$. Suppose further $f$ is a real polynomial in $n$ variable s.t. $f$ is irreducible over $C$. $\textbf{Q:}$ Now consider $I(V(f))\...
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32 views

Closed embedding in flat topology

Let $j:Y \to X$ be a closed embedding of schemes. Then we know there is an exact sequence in the Zariski topology $0 \to I \to \mathcal{O}_X \to j_{*} \mathcal{O}_Y \to 0$. Now consider the sheaf $\...
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1answer
33 views

If $d\mid e$, does the ideal $(x^d-1, 1+x+\cdots+x^{e-1})\subset\mathbb{Z}[x]$ contain $1+x+\cdots x^{d-1}$?

If $d\mid e$, does the ideal $(x^d-1, 1+x+\cdots+x^{e-1})\subset\mathbb{Z}[x]$ contain $1+x+\cdots x^{d-1}$? For an integer $n$, let $\chi_n(X) := 1+X+\cdots+X^{n-1}$. Certainly $\chi_d(x)$ divides ...
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2answers
97 views

Nullstellensatz and Proper Ideals [on hold]

Let $I \subset \mathbb{C}[x_1,x_2]$ be a proper ideal. How does it follow from Hilbert's Nullstellensatz, that there must exist $z_1,z_2 \in \mathbb{C}$ such that $I \subset \left\langle x_1-z_1,x_2-...
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1answer
37 views

About the definition of subring

Reading Atiyah-MacDonald: Introduction to Commutative Algebra, I found the following definition of subring: A subset $S$ of a ring $A$ is a subring of $A$ if $S$ is closed under addition and ...
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1answer
28 views

R with different Jacobson radical and nilradical (both non-zero)

I have seen a post in MathStack with $R=\mathbb{Z}[x]$, $p=(x^2+1)$ and $m=(x^2+1,2)$. I can not understand this example. Can you help me to understand it? Or may be giving any other example? I ...
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33 views

maximal and largest sub-field of $S^{-1}\mathbb{C}[x,y]$

Consider $S^{-1}\mathbb{C}[x,y]$ such that $S$ is generated by $\{x-a | a \in \mathbb{C}\} \cup \{y-a | a \in \mathbb{C}\}$. With the equivalence relation $[f,s] \sim [f',s']$ iff $\exists s'' \in S$ ...
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1answer
24 views

Valuation ring's principal ideals

Let $V$ be a valuation ring. Then any two principal ideals $A_1$ and $A_2$ of $V$ are ordered by inclusion. I need a proof for this lemma and I don't know how to start.
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1answer
28 views
+50

Question about a proof for extending a local homomorphism to a morphism of schemes

Let $X,Y$ be $S$-schemes, let $x\in X$ and $y\in Y$ be points over $s\in S$ and let $\mathcal O_{Y,y}\to\mathcal O_{X,x}$ be a local $\mathcal O_{S,s}$-homomorphism. If $Y$ is locally of finite ...
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26 views

zero-divisors of a ring constitute an ideal

I want to know if "zero-divisors of a ring constitute an ideal iff each pair of zero-divisors of the ring has a nonzero annihilator?" the crucial point for zero-divisors of a ring to constitute an ...
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1answer
25 views

Sheaves of abelian groups and tensor product

Let $X$ be a topological space and $\mathcal{F}$ and $\mathcal{G}$ two sheaves of abelian groups. Now let me define a presheaf $\mathcal{F} \otimes_{\mathbb{Z}} \mathcal{G}$ such that $\mathcal{F} \...
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2answers
44 views

Cutting down an Ideal

Let $A \subset B$ two commutative rings and $I \subset A$ an Ideal. I shall write $I B$ for the Ideal in $B$ generated by $I$. Now, is it always true that $I = A \cap I B$ ? We used something like ...
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2answers
51 views

Constructing a ring with a given spectrum

A DVR necessarily has spectrum $\{ 0, \mathfrak{m} \}$, but a DVR is also necessarily noetherian. Can we find an example of a non-noetherian (thus non DVR) ring, say $R$, with the spectrum $\{ 0, \...
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Relation between the split extension and nonsplit extensions.

Suppose $A$ is an algebra over $\mathbb{C}$. Let $M$ and $N$ be $A$-modules with $Ext^{1}(M,N) \neq 0 \neq Ext^{1}(N,M)$. By $E\in Ext^1(N,M)$, I mean we have a short exact sequence of the form $$ 0 ...
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22 views

An example of morphism not of finite type

This is example 3.2.2 of Chapter II of Hartshorne's algebraic geometry. If P is a point of a variety V, with local ring $O_P$, then $Spec O_P$ in general of finite type. I did not see an example right ...
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2answers
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Completion (construction Atiyah MacDonald chapter 10)

Following Atiyah MacDonald (Chapter 10: Completions), let $G$ be a topological abelian group. We assume that $0 \in G$ has a fundamental system of neighborhoods consisting of subgroups $G = G_0 \...
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Writing out a maximal sub-field of $R$

Consider the ring $R = S^{-1} \mathbb{C}[x,y]$ where $S$ is the multiplicative set generated by $$\left\{x - a \mid a \in \mathbb{C} \right\} \cup \left\{ y - a \mid a \in \mathbb{C} \right\}. $$ I ...
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35 views

Do we have $\ker (f)_{\mathfrak p}\simeq \ker (g)$?

Let $A,B$ be commutative rings with identity and $f:A\to B$ a homomorphism of rings. For any prime ideal $\mathfrak q$ of $B$, denote $f^{-1}(\mathfrak q)$ by $\mathfrak p$, then $f$ induces the ...
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1answer
65 views

How to show the canonical homomorphism $A_{\mathfrak p}\to B_{\mathfrak q}$ is injective?

Let $A,B$ be commutative rings with identity and $f:A\to B$ an injective homomorphism of rings. For any prime ideal $\mathfrak q$ of $B$, denote $f^{-1}(\mathfrak q)$ by $\mathfrak p$, how to show ...
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21 views

Show that $M$ can be generated by the maximal ideals in a Noetherian semilocal ring.

Let $R$ be a Noetherian semilocal ring and $M$ a finite $R$-module. Write $n=\max\{\mu_{R_m}(M_m)|m\in m-\text{Spec}(R)\}$. Show that $M$ can be generated by $n$ elements. Doesn't this come directly ...
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30 views

Is the ideal $\langle x_1^2-1, x_2^2-1, x_3^2-1, (x_3-x_1)(x_3-x_2) \rangle$ homogeneous? [on hold]

Let $R=\mathbb{C}[x_1,x_2,x_3]$. Is the ideal $I=\langle x_1^2-1, x_2^2-1, x_3^2-1, (x_3-x_1)(x_3-x_2) \rangle$ a homogeneous ideal of $R$? I checked on Macaulay2 that we can still compute the ...
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about flatness of module

I need help in (iii) implies (iv) in this why tor (M,IN)=0 and tor(M,N/IN)=0 ;and (v) implies (1). In (v) implies (i) I understand the proof until the use of artin-rees lemma but after that I don't ...
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0answers
27 views

Is $\overline{\mathbb{Z}}$ noetherian in $\overline{\mathbb{Q}}$? [duplicate]

Consider the algebraic closure of its fraction field $K=\overline{\operatorname{Frac}(\mathbb{Z})}=\overline{\mathbb{Q}}$. Is the integral closure $\overline{\mathbb{Z}}$ in $K$ noetherian? I do not ...
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45 views

Elements of $R=S^{-1} \mathbb{C}[ x,y ]$

I would like to understand how are the elements of $R=S^{-1} \mathbb{C}[ x,y ]$ where S is the multiplicative set generated by $\{x-a | a \in \mathbb{C} \} \cup \{y-a | a \in \mathbb{C} \}$. I would ...
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0answers
29 views

Uniqueness in Weierstraß p-adic preparation theorem

I have given the Weierstraß p-adic preparation theorem. It is stated as follows: Let $f=a_0 + a_1T + ... \in \mathbb{Z}_p[[T]]$ for a prime $p$ such that $p \mid a_0,...,a_{n-1}$ and $p\not\mid a_n$. ...
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35 views

Is the annihilator of a minimal prime always principal in a reduced ring?

$\DeclareMathOperator{\Ann}{Ann}$Let $R$ be a (one-dimensional) reduced ring and let $I$ be a non-zero minimal prime ideal of $R$. I am looking for an example such that $\Ann(I)$ is not principal. ...
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34 views

Does $\operatorname{Ann}(d) = \operatorname{Ann}(b)$ together with $d \in \operatorname{Rad}(bR)$ imply $d \in bR$?

$\DeclareMathOperator{\Ann}{Ann}\DeclareMathOperator{\Rad}{Rad}$Let $R$ be a noetherian, reduced ring which is not an integral domain (and may also not be factorial). Let $P$ be a minimal prime of ...
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Showing that $A[x] \cap A[x^{-1}] $ is an integral extension of $A$ when $x$ is invertible and $x$ belongs to a bigger ring containing $A$ [duplicate]

I have a problem- $A, B$ are rings such that $A \subset B$ and $x$ is an invertible element in $B$. I have to show that that the ring $T = A[x] \cap A[x^{-1}]$ is an integral extension of $A$...
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1answer
59 views

Calculation of valuation ring of a valuation associated to a blowup

Let $\mathfrak{m} = (x,y) \subset k[x,y]$. Then the valuation $v$ of $k(x,y)$ associated to the exceptional divisor of the blowup should be defined by $$v(f) = \mathrm{sup}(n|f \in \mathfrak{m}^n), f\...
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1answer
45 views

An isomorphism about localisations

Let $R=\mathbb{R}[x,y]/(xy)$ and $P\subset R$ the ideal generated by $x-1$. Show that $R_P\cong \mathbb{R}[x]_{(x-1)}$. My attempt: first I tried to use the isomorphism between $\mathbb{R}[x,y]/(xy)$ ...
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1answer
22 views

Existence of a subring of a Noetherian ring with no finite bases in it.

Consider complex polynomial ring $R=\mathbb{C}[x_1,x_2,x_3,x_4]$. By Hilbert's basis theorem it's a Noetherian ring. So I am wondering whether there exists an ideal that has no finite basis in itself. ...
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1answer
26 views

Injective resolution of $\mathbb Z / p$ but resulting objects not injective?

I'm trying to find an injective resolution for the $\mathbb Z$-module $\mathbb Z/p$ where $p$ is a prime. I've come across the result that an injective resoltion for a PID $R$ is of the form $$ 0 \...
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0answers
42 views

Showing some element is a zero divisor

Let $R=R_1\oplus R_2$, where each $R_i$ is a commutative ring with unity. Let $(S,\eta)$ be a local subring of $R$. Let $\pi_1$ be the projection of $R$ onto $R_1$. It is also given that $\pi_1|_S$ is ...
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20 views

Mutual Co-primeness of two multivariate polynomials

We know that in case of integers for two integers $m$ and $n$ are co-prime means we have an identity $pm+qn$=1,for some appropriate integers $p$ and $q$.I think that this happens for any Euclidean ...
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1answer
24 views

Dual of a differential graded module

Let $(M, d_M)$ be a differential $\mathbb{Z}$-graded module over a differential graded algebra (over a field) $(A, d_A)$. I am wondering if there is a canonical way of looking at the dual $\hom_A(M,A)...
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2answers
36 views

Localizing the ring of integers of a number field to get a PID

Let $K$ be a number field with ring of integers $R$. Let $I_1,\ldots, I_h$ be (integral) ideals representing generators for the class group of $K$, and let $S$ be the set of prime ideals (a.k.a. ...
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1answer
47 views

Is $R/bR$ reduced if $R$ is reduced and $\operatorname{Ann}(b)$ is a minimal prime of $R$?

Let $R$ be a noetherian, reduced ring. Let $P = \operatorname{Ann}_R(b)$ with $b \in R$ be a minimal prime of $R$. Does this imply that $R/bR$ is reduced? Of course, the property in question is ...
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1answer
23 views

If $N$ is a pure submodule of $M$ and $\text{ann}(x+N)=(d)$, prove that $x$ can be chosen in its coset $x+N$ so that $\text{ann} x=(d)$.

Let $D$ be a principal ideal domain (PID) and let $M$ be a $D$-module. A submodule $N$ of $M$ is said to be pure in $M$ if $N\cap rM=rN$ for all $r\in D$. If $N$ is a pure submodule of $M$ and $\text{...
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28 views

Is the complex polynomial ring finitely generated? [closed]

Whether the complex polynomial ring is a Notherian ring? If it's true, how can we find its generators?
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47 views

Picard group of projective line

Let $R$ be some $k$-algebra. Consider the projective line over $R$, $\mathbb{P}^1_R$. What is $Pic(\mathbb{P}^1_R)$?
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20 views

Kernel of a linear map between polynomial rings

Let $k[a, b, c, d]^{\Sigma_{2}}$ be a ring of 4-variable polynomials which satisfies $f(-a, -b, -c, -d) = f(a, b, c, d)$, i.e. invariant under the $\Sigma_{2} = \mathbb{Z}/2$-action on $k[a, b,c,d]$ ...
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0answers
57 views

A product of maximal ideals is zero

Show that the zero ideal is a product of maximal ideals (not necessarily distinct) in the ring $k[x,y,z]/(x(x-1),y^2,z^3)$. I tried using Nullstellensatz and then the 4th Isomorphism Theorem, to get ...
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0answers
22 views

Reference of a theorem in EGA about faithfully flat ring changes

Is there an english (or german) reference for the proposition (2.5.8) that can be found in EGA, IV? The proposition is more or less the following, simplified to my application Proposition Let $A$ be ...
4
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1answer
46 views

Is the power series ring over a polynomial ring the same as the polynomial ring over the power series ring?

Let $R$ be a commutative ring with $1.$ Then consider the power series ring over a polynomial ring as $R[X][[Y]]$ and the polynomial ring over a power series ring $R[[Y]][X].$ Are these two objects ...
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1answer
28 views

Localization of Formal Power Series

Quick question. Is the following statement correct? Let $F$ be a field and $F[[x]]$ be a ring of its formal power series in the indeterminate $x$. Then localization of $F$ at $S = \{x^n\;|\;n>0\}$ ...
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0answers
33 views

Non-invertible elements in an algebra which do not belong to the kernel of any character

Is there a commutative unital algebra $A$ over $\mathbb C$ such that for some $f\in A$, $\chi(f)\not=0$ for every (non-zero) homomorphism $\chi:A\to\mathbb C$, but for which $f$ is not ...