# Tagged Questions

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363 views

### Submodule embeddable in a finitely generated module

I have a terminology question. For a commutative ring $A$ (not necessarily Noetherian), $A$-modules that are isomorphic to an $A$-submodule of a finitely generated $A$-module form a fairly good class ...

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43 views

### $I$ is principal if and only if $ID_M$ is principal for all $M \in Max(D)$

In a semmi-quasi local domain $D$ ( i.e. $D$ has finitely many maximal ideals),
an ideal $I$ is principal if and only if $ID_M$ is principal for all $M \in Max(D)$.
[See Comm. Rings by Kaplansky, ...

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votes

**1**answer

162 views

### $ mult(R/I) = d_1 \cdots d_r \quad \Rightarrow \quad f_1,\dots,f_r \quad \text{is a $R$-regular sequence?}$

We define multiplicity of a module M of dimension $d>0$ as $$mult(M) := lc (P_M) (d-1)!$$ where $P_M$ denotes the Hilbert polynomial of M. Equivalently, we have $mult(M) = Q_M(1)$, where $HP_M (z) ...

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**1**answer

86 views

### Isomorphic quotient of a Module over Noetherian commutative algebra [closed]

I have a nice solution to the following problem and I thought of writing a paper about it but beforehand, I wanted to ask the problem here to see if this is an easy problem and if you people can solve ...

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73 views

### Recovering fractional ideals from ideals

Let $R$ be a Noetherian domain; let ${I_{(i)}}$ be a set of fractional ideals in $K$, the fraction field of $R$, indexed by a lattice such that $I_{(i)} I_{(j)} = I_{(i + j)}$. Let $J_{(i)} = I_{(i)} ...

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**0**answers

68 views

### Completion of Bezout Domain a Bezout Domain?

Let $R$ be a Bezout domain, and $I$ any ideal inside of $R$. Is the $I$-adic completion
$$ \varprojlim_i R/I^i $$
necessarily a Bezout domain? If not, what conditions (on $R$ or $I$) might ensure that ...

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**2**answers

324 views

### What is the probability that a random sequence of polynomials is regular?

Let $k$ be a finite field or a field with a height function, such as a number field.
Consider the ring $k[[x_1,\dots, x_n]]$ and let $\mathfrak{m}$ be its maximal ideal.
What is the asymptotic ...

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**2**answers

258 views

### In this special situation, does $M \otimes B=0$ imply $M=0$?

Let $\Phi:A \rightarrow B$ be a flat morphism of commutative rings. Let $f \in A$, not a unit and $A/fA \cong B/fB$ induced by $\Phi$.
Let $M$ be an $A_f$-module. Is it true that $M \otimes_A B = 0 ...

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31 views

### Bigraded analogue of Ratliff-Rush closure filtration

Consider the filtration $\lbrace{I^rJ^s}\rbrace_{r,s\in\mathbb{Z}}.$
What will be the bigraded analogue of Ratliff-Rush closure filtration $\tilde{{I}^n}=\cup_{k\geq1}({I}^{n+k}:{I}^k)$?
Will it be ...

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49 views

### Depth of multigraded modules

Can any one please give me some references on depth of multigraded module over a standard multigraded ring.

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**1**answer

130 views

### Idempotent fractional ideals of a Noetherian domain

Let $R$ be a commutative Noetherian domain, $K$ its fraction field, and $J$ a fractional ideal (i.e. a finitely generated sub-$R$-module of $K$) such that $J^2=J$. Is it true that $J=0$ or $J=R$? If ...

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**1**answer

523 views

### Algebraic Closure of a Ring is Not a Ring?

I'm trying to motivate the notion of integrality in a ring extension. It seems that the following would be a good motivation, because it would show that the notion of algebraic elements over a ring is ...

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**1**answer

163 views

### reduction of an admissible filtration

Let $(R,m)$ be a local ring and $I$ an $m$-primary ideal of $R.$ $\lbrace I_n\rbrace_{n\in\mathbb{Z} }$ is called $I$-admissible filtration
1) if $m\geq n$ then $I_m\subset I_n.$
2) for all $m,n,$ ...

**0**

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**1**answer

95 views

### Relation between local cohomology and koszul cohomology of multigraded ring

Let $R$ be a ${\mathbb {Z}}^k$-graded Noetherian ring, $J=(x,y)$ an ideal of $R$ where $x,y\in R_{(1,\ldots,1)}.$ Is this following true $$H_{J}^i(R)=\underset{n}\varinjlim{H^i((x^n,y^n),R)},$$ where ...

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238 views

### intersection of finitely many maximal ideals

For what commutative rings with infinitely many maximal ideals we can say that the intersection of any combination of finitely many maximal ideals is not zero?
Obviously it holds for Dedekind domains ...

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**2**answers

428 views

### An identity in an arbitrary commutative ring

This fact might be either trivial, wrong, or well known.
Let $R$ be a commutative ring. Let $u_1,\dots,u_{s-1},u_s\in R$ and $m,M\in R$. Let us assume that $m,M$ satisfy
$$(m-u_1) \dots ...

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**2**answers

242 views

### Tensor powers of an algebra all isomorphic

Let $k$ be a commutative ring with total quotient ring $K$, and let $A$ be a commutative $k$-algebra such that the multiplication map $A \otimes_k A \longrightarrow A$ is an isomorphism.
EDIT: Assume ...

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221 views

### Condition for a local ring whose maximal ideal is principal to be Noetherian

The statement "a local ring whose maximal ideal is principal is Noetherian" is (I think) false. The ring of germs about $0$ of $C^\infty$ functions on the real line seems to be a counterexample since ...

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**0**answers

64 views

### Graded Betti Numbers of a Graded Ideal with Linear Quotients

Exercise 8.8 in Monomial Ideals by Herzog and Hibi:
Let $I\subset S=K[x_{1},...,x_{n}]$ be a graded ideal which has linear quotients with respect to a homogeneous system of generators ...

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38 views

### Graded Betti Numbers of a Stable Monomial Ideal

Exercise 8.8 in Monomial Ideals by Herzog and Hibi:
Let $I\subset S=K[x_{1},...,x_{n}]$ be a stable monomial ideal with $G(I)=\{u_{1},...,u_{m}\}$ and such that for $i<j$, either ...

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**1**answer

167 views

### Checking flatness using radical ideals

Let $R$ be a commutative ring and $M$ a not necessary finitely presented $R$-module. I am looking for a prove or a counterexample to the following statement: $M$ is flat as an $R$-module if and only ...

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**0**answers

97 views

### Universally catenary and all its formal fibers over minimal members are Cohen-Macaulay but it has a nonCohen-Macaulay formal fiber

Please help me to find a Noetherian local ring $R$ such that: $R$ is universally catenary and all its formal fibers over minimal members of $Spec(R)$ are Cohen-Macaulay but $R$ has a nonCohen-Macaulay ...

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159 views

### Square of primary ideals

Is there any example of a $P$-primary ideal $I$ in a noetherian domain $R$ such that $I^2=PI \not=P^2$?

**0**

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**1**answer

148 views

### Depth of polynomial ring $S=\Bbb{R}[x_1,x_2,x_3,…,x_n,…]$

Consider the polynomial ring of countable variables with coefficients in the real numbers, i.e, $S=\Bbb{R}[x_1,x_2,x_3,...,x_n,...]$. My Question is about the depth of this ring.
Question: Could we ...

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**3**answers

185 views

### Subset of Spec(A) realized as inverse image of some Spec(B)

Let $A$ be a (commutative) ring and $U$ be an arbitrary subset of $\mathrm{Spec}(A)$. Do there exist a ring $B$ and a ring homomorphism $\varphi\colon A\to B$ such that ...

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**1**answer

183 views

### chain of prime ideals in polynomial ring $S=\Bbb{R}[x_1,x_2,…,x_n,…]$

Consider the polynomial ring of countable variables with coefficients in the real numbers, i.e, $S=\Bbb{R}[x_1,x_2,x_3,...,x_n,...]$. Is the following question true?
Question: Is there any maximal ...

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172 views

### Is the generation of rings by their units a question in K-theory?

Susan's question When can number rings be spanned (as $\mathbb{Z}$-modules) by units? smells like an algebraic K-theory question in disguise. I'll reformulate the question first:
Given an integral ...

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61 views

### Hilbert Regularity in relation to degree of generators

Suppose we look at the $\mathbb {C}$ module $\mathbb{C}[x_1,\dots,x_n]=\oplus{S_i}$ where $S_i$ are the polynomials of degree $i$. Then we look at a subring $R$ (also a $\mathbb {C} $ module) ...

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**1**answer

317 views

### Is an ideal generated by multilinear, irreducible, homogeneous polynomials of different degrees always radical?

I asked this question on math.se and someone even put a bounty on it, yet there was no answer. Hence, I am asking here. Assume $\Bbbk$ to be a field of characteristic zero.
Definition. A ...

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52 views

### lift isomorphic in a sufficiently thick fiber

Let $P$ and $P'$ two polynomials in $k[[\pi]][t]$ for an algebraically closed field $k$ and let $A=k[[\pi]]$.
We consider $ X'=Spec (A[t]/(P'))$ and $X=Spec (A[t]/(P)$.
Let $d=val(\Delta(P))$ where ...

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**1**answer

86 views

### on smoothness of morphisms on an artinian base

Let $A$, $B$ two smooth $R$-algebras of finite type for a artinian local ring $R$.
Let $I$ an ideal such that $I^{2}=0$ and $\bar{R}=R/I$.
We assume that the map $Spec B/IB\rightarrow Spec A/IA$ is ...

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**1**answer

166 views

### elementary question on a completion of a ring

Let $k$ a field, and $k[\epsilon]=k[X]/(X^{2})$ , what is the completion of the ring $k[\epsilon][t]$ with respect to the ideal $(t^{2}+\epsilon)$?

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**1**answer

209 views

### Symmetric algebra of an ideal and syzygies

Let $(R,\mathfrak{m})$ be a Cohen Macaulay local ring and $I=(a_1,\ldots,a_g,a_{g+1})$ an almost complete intersection ideal of codimension $g.$ Let $R^k\longrightarrow R^{g+1}\longrightarrow ...

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**3**answers

420 views

### Idempotent polynomials

Let $R$ be a commutative ring with identity and let $f \in R[x]$. There are well known characterizations for $f$ to be a nilpotent element of $R[x]$ or to have a multiplicative inverse in $R[x]$. Is ...

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**2**answers

526 views

### Example of a reduced ring whose completion is not reduced

Let $(R,\mathfrak{m})$ be a local Noetherian ring. Give an example such that R is reduced but $\mathfrak{m}$-adic completion of R is not reduced.

**1**

vote

**1**answer

122 views

### Finding reducible polynomials with restricted factors

Given $f(x),g(x) \in \mathbb{Z}[x]$, two irreducible polynomials, is there a polynomial $h(x) \in \mathbb{Z}[x]$ coprime to $f(x)$ such that $f(x) + g(x)h(x)$ is reducible over $\mathbb{Z}[x]$ with ...

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**2**answers

202 views

### Decomposition of a quotient module

Let $R=k[v,x,y,z]/I$, with $I=\langle v^2,z^2,xy,vx+xz,vy+yz,vx+y^2,vy-x^2\rangle$,and let
$f:R^2 \rightarrow R^2$ denote the map given by the matrix
$$M=\begin{pmatrix}
v & y \\
x & z
...

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116 views

### Ring algebraically closed in its completion.

First I would like to be clear about the definition, which I am having trouble finding.
What does: The local ring $A$ is algebraically closed in $B\supset A$. (e.g. for $B:=\hat{A}$, the completion ...

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**1**answer

162 views

### for which truth-operations f can f-membership in a prime ideal be represented by a polynomial?

Nota bene: all rings are supposed commutative with $1$ and all ring homomorphism should be unital.
Let $B$ be the set of truth values {True, False}. For a formula $\phi$ denote by $[\phi]\in B$ it's ...

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**2**answers

327 views

### Surjectivity of the natural map of injective module to its localization

Hello all,
The lemma 3.3 page 214 in Hartshorne Algebraic Geometry book states:" If $I$ is an injective module over a Noetherian ring $A$. Then for any $f\in A$, the natural map of $I$ to its ...

**0**

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**0**answers

69 views

### Example of a ring whose minimals are annihilators of idempotents?

I'm looking for examplesâ€ of rings with the property that for each
$P={\rm Ann}_R(a)\in{\rm Min}(R)$ then $a\in R$ is idempotent (ie $a^2=a$)
â€ other than domains!

**6**

votes

**1**answer

411 views

### Are roots of transcendental elements transcendental?

This looks extremely easy, but then again it's late at night...
Let $k$ be a commutative ring with unity. An element $a$ of a $k$-algebra $A$ is said to be transcendental over $k$ if and only if ...

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votes

**1**answer

144 views

### Relation between $H^i_I(-)$ and $H^i_J(-)$ when $I\subset J$

What is the relation between $H^i_I(-)$ and $H^i_J(-)$ (cohomological functors) when $I\subset J$ are ideals of a (local) noetherian ring?

**1**

vote

**1**answer

115 views

### An example of a ring $R$ with the property that for each $P=Ann_R(r)\in {\rm Min}(R)$ we have $Ann_R(P)=Rr$.

I'm looking for an example of a commutative (preferably local) ring $R$ such that ${\rm dim}R>0$ and $R$ has the property that for each $P=Ann_R(r)\in {\rm Min}(R)$ we have $Ann_R(P)=Rr$.
This ...

**0**

votes

**1**answer

75 views

### Explicit representation of $R[\frac{x}{y}]$ where $x, y\in R$ for non-Euclidean PIDs $R$?

It's a fact proven by Pendleton, Gilmer, and Ohm (as an obvious corollary of their work, anyways) that PIDs are QR-domains, meaning every overring (ring between the domain and the quotient field) is a ...

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votes

**3**answers

199 views

### When does “second annihilator” of a (principal) ideal equal the ideal itself , ie $Ann_R(Ann_R(r))=Rr$?

Suppose that $R$ is a (local) ring and $r\in R$. When do the equations $Ann_R(Ann_R(r))=Rr$ or $\sqrt{Ann_R(Ann_R(r))}=\sqrt{Rr}$ hold?
I already know that it holds for Artinian Gorenstein rings (due ...

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**0**answers

95 views

### Number of generators of colon and intersection ideals of two finitely generated ideals in a Prufer domain

Hi!
I'm wanting to see why the following is true: Given 2 finitely generated ideals $B$ and $C$ in a Prufer domain $D$ with bases of $n$ and $m$ generators respectively, $B\cap C$ has a basis of ...

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**1**answer

163 views

### Example of an overring of an integral domain which is not a ring of quotients?

Hi!
I'm trying to make headway on a question for my undergraduate honors thesis, specifically the question of which rings of integer-valued polynomials if any satisfy the QR-property; that is, the ...

**0**

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**1**answer

154 views

### lattice of subalgebras of a finite commutative algebra

(I) Suppose A is a finite commutative local algebra. Must every lattice of local subalgebras of A be a distributive lattice ?
By a subalgebra of A we mean an algebra contained in A that shares the ...

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**1**answer

492 views

### “thematic” algebras

I scoured what I could in the literature but I have yet to find the information that should be out there. Consider the property
(P1) Every local subalgebra can be embedded in a local ideal ...