Questions tagged [commutative-algebra]
Questions about commutative rings, their ideals, and their modules.
10,046 questions
1
vote
0answers
18 views
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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^{\...
0
votes
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 ...
1
vote
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))\...
2
votes
0answers
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 $\...
2
votes
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 ...
-2
votes
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-...
0
votes
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 ...
1
vote
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 ...
0
votes
0answers
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 $\exists s'' \in S$ ...
0
votes
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.
1
vote
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 ...
1
vote
0answers
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 ...
0
votes
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} \...
2
votes
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 ...
3
votes
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, \...
1
vote
0answers
32 views
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
$$ 0 ...
1
vote
0answers
22 views
An example of morphism not of finite type
This is example 3.2.2 of Chapter II of Hartshorne's algebraic geometry. If P is a point of a variety V, with local ring $O_P$, then $Spec O_P$ in general of finite type. I did not see an example right ...
0
votes
2answers
28 views
Completion (construction Atiyah MacDonald chapter 10)
Following Atiyah MacDonald (Chapter 10: Completions), let $G$ be a topological abelian group. We assume that $0 \in G$ has a fundamental system of neighborhoods consisting of subgroups $G = G_0 \...
1
vote
0answers
47 views
Writing out a maximal sub-field of $R$
Consider the ring $R = S^{-1} \mathbb{C}[x,y]$ where $S$ is the multiplicative set generated by $$\left\{x - a \mid a \in \mathbb{C} \right\} \cup \left\{ y - a \mid a \in \mathbb{C} \right\}. $$
I ...
0
votes
0answers
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 ...
1
vote
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 ...
-1
votes
0answers
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 ...
-1
votes
0answers
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 ...
0
votes
0answers
21 views
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 ...
0
votes
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 ...
3
votes
0answers
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 ...
1
vote
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$. ...
0
votes
0answers
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.
...
0
votes
0answers
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 ...
1
vote
0answers
17 views
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$...
3
votes
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\...
2
votes
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 $\mathbb{R}[x,y]/(xy)$ ...
0
votes
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. ...
0
votes
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 \...
2
votes
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 ...
0
votes
0answers
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 ...
2
votes
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)...
2
votes
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. ...
1
vote
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 ...
1
vote
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{...
0
votes
0answers
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?
0
votes
0answers
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 $Pic(\mathbb{P}^1_R)$?
0
votes
0answers
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 $k[a, b,c,d]$ ...
1
vote
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 ...
0
votes
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
votes
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 ...
1
vote
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 $S = \{x^n\;|\;n>0\}$ ...
1
vote
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 ...
