The commutative-rings tag has no usage guidance.

**1**

vote

**0**answers

140 views

### local rings with finite type maximal ideal

Let $A$ be a local ring with a maximal ideal $\mathfrak{m}$ finitely generated (not principal).
Is there a sufficient condition for $A$ to be noetherian?
For example, we know that the completion ...

**6**

votes

**0**answers

122 views

### flatness and derived completion

Let $A$ be a local ring of maximal ideal $\mathfrak{m}$. Let $\hat{A}$ be its completion.
If $A$ is noetherian , then we know that $A\rightarrow\hat{A}$ is faithfully flat.
If $A$ is not noetherian, ...

**1**

vote

**2**answers

164 views

### if $R$ is Noetherian local with a finite module of finite injective dimension and if “?” , then $R$ is “Gorenstein”

I know that if $R$ is Noetherian local with a finite module of finite injective dimension, then $R$ is Cohen-Macaulay.
Can one add assumptions on $M$, so that $R$ be Gorenstein or Complete ...

**0**

votes

**0**answers

48 views

### normality of truncated arc space

Let $X=Spec(A)$, with $A$ a normal $k$-algebra of finite type, $k$ is a field.
For any integer $n$, let $X(k[t]/(t^{n}))$ the $n$-th truncated arc space, is it also normal?
Same question for ...

**9**

votes

**0**answers

129 views

### Weierstrass division theorem for henselian rings

Let $A$ be an henselian local noetherian ring. There is an old result of Lafon ("Anneaux henséliens et théorème de préparation" (1967)), which says that if $A$ is analytically normal and of ...

**3**

votes

**3**answers

205 views

### canonical module can be identified with an ideal. how can one reach that ideal?

Let $R[[X,Y,Z]]/(X,Y)\cap (Y,Z)\cap(X,Z)$. then $R$ is Cohen-Macaulay ring and has a canonical
module, $K$. By Proposition 3.3.18 of Bruns_Herzog, $K$ can be identified with an ideal in $R$.
So we ...

**1**

vote

**1**answer

130 views

### Graded version of Baer's Criterion

Baer's Criterion for injectiveness of modules says: "An $R$-module $E$ is injective iff for all ideals $I$ of $R$, every homomorphism $f\colon I \to E$ can be extended to $R$." I wonder if there is a ...

**0**

votes

**1**answer

120 views

### Canonical module of rees algebra

[Example 4.27, Integral Closure, Rees Algebras, Multiplicities, Algorithms] by Vasconcelos, says that if $I=(f_1,\ldots,f_g)$ is an ideal generated by a regular sequence with $g\ge 2$ then the ...

**2**

votes

**1**answer

104 views

### Matching power series to infinity

As pointed out by Makoto, on this question about power series rings and the axiom of choice, an idea I had needed the axiom of dependent choice to work. However, the construction raises another ...

**2**

votes

**1**answer

205 views

### finiteness dimension

$R$ is a local Noetherian ring. $f_I(M)$, the finiteness dimension of a module $M$ relative to $I$, is defined in ...

**0**

votes

**1**answer

142 views

### if $ \lambda (I)= \dim R$, can one claim that $I$ is an $m$-primary ideal?

definition from Bruns-Herzog:
It is easy to see that if $I$ is a $m$-primary ideal of $R$ then $ \lambda (I)= \dim R$. I wonder if the converse is true:
if $ \lambda (I)= \dim R$, can one ...

**1**

vote

**1**answer

255 views

### intuitive interpretation of analytic spread

I am studying analytic spreads from Bruns-Herzog's book. The definition is clear but calculation of the analytic spread of an ideal is hard for me in practice. I wonder if it is hard for you too.
...

**1**

vote

**0**answers

144 views

### On Prüfer domains

Is there any Prüfer domain $R$ that has a prime ideal $P$ that is not finitely generated but $xP$ is subset of a finitely generated ideal $I$,for some $x$ in $R-P$ and $I$⊂$P$?

**5**

votes

**2**answers

240 views

### Integral domains with totally ordered spectra

In my research I ended up trying to prove some properties of integral domains such that their spectrum is a totally ordered poset. Are there some nice (ubiqitous/natural) examples of such domains, ...

**0**

votes

**1**answer

113 views

### Bounded Index of Nilpotency of $R[x]$

A ring $R$ is called with bounded index (of nilpotency) $n$ if $n$ is the smallest natural number such that $a^n=0$ for all nilpotent $a \in R$.
Now let $R$ be a commutatitve ring with bounded index ...

**1**

vote

**0**answers

65 views

### Ideals in a reduced ring

In a finite reduced commutative ring, is every minimal prime ideal is generated by an idempotent?
The number of idempotent elements and the number of minimal prime ideals are same.

**1**

vote

**0**answers

71 views

### Projective dimension of a sub-ideal

Let $\mathbf{k}$ be a field, and let $S=\mathbf{k}[x_1,x_2,\ldots,x_n]$. Let $I\subset J$ be finitely generated monomial ideals in $S$. Is it true that the projective dimension of $I$ is either ...

**1**

vote

**1**answer

190 views

### Quotients and radicals

Let $I, J$ be ideals in a commutative ring with identity $R$. Define the quotient ideal $(I : J)$ by $$(I : J)=\{x\in R : xJ\subseteq I\}.$$
Define the radical $r(A)$, of an ideal $A$ of $R$ by
...

**0**

votes

**1**answer

249 views

### Uniform Artin-Rees

The Artin-Rees lemma states that if $R$ is a Noetherian ring, $I \subseteq R$ is an ideal and $N \subseteq M$ are finitely generated $R$-modules, then there exists an integer $k$ such that for every ...

**1**

vote

**0**answers

191 views

### Rings of algebraic integers as quotients of polynomial rings

The ring of integers $\mathcal{O}_K$ of a number field $K$ is always isomorphic to some ring of the form $\mathbb{Z}[x_1, ..., x_r]/\mathfrak{p}$, where $\mathfrak{p} \subset \mathbb{Z}[x_1, ..., ...

**1**

vote

**0**answers

458 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 ...

**0**

votes

**1**answer

95 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, ...

**5**

votes

**1**answer

242 views

### Does $ \text{mult}(R / I) = d_{1} \cdots d_{r} $ imply that $ (f_{1},\ldots,f_{r}) $ is an $ R $-regular sequence?

We define the multiplicity of an $ R $-module $ M $ of dimension $ d > 0 $ to be
$$
\text{mult}(M) \stackrel{\text{df}}{=} \text{LC}(P_{M}) \cdot (d - 1)!,
$$
where $ P_{M} $ denotes the Hilbert ...

**-3**

votes

**1**answer

131 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 ...

**1**

vote

**0**answers

85 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)} ...

**2**

votes

**0**answers

78 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 ...

**7**

votes

**2**answers

342 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 ...

**1**

vote

**0**answers

68 views

### Irreducibility of a certain matrix variety

Let $R=\mathbb{Q}[x_{i,j}\,:\, 1\leq i,j\leq n]$. Let $M$ be the $n\times n$ matrix $(x_{i.j})$. Let $\chi(M)$ be the characteristic polynomial of $M$. Finally, let $I$ be the ideal of $R$ ...

**1**

vote

**2**answers

301 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 ...

**2**

votes

**1**answer

144 views

### Nilradical and Newton's identities

Let $R$ be a commutative ring with unity such that $n!$ is not a zero-divisor. Let $s_1=\sigma_1,s_2,s_3\cdots$ and $\sigma_1,\sigma_2\cdots$, (convention: if $k>n$, then $\sigma_k=0$) be elements ...

**5**

votes

**1**answer

187 views

### Units of $\mathbf Z[X,Y]/(P(X,Y))$

Let $P(X,Y)\in \mathbf Z[X,Y]$ be an irreducible polynomial and let $A$ denote the quotient ring $\mathbf Z[X,Y]/(P)$.
What is known about the group of units of $A$?
It's not even clear to me that ...

**3**

votes

**1**answer

159 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 ...

**8**

votes

**1**answer

715 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 ...

**2**

votes

**0**answers

163 views

### Domains with prime ideal theorems

Let $D$ be a domain, and for prime ideals $\frak P$ of $D$ the norm is $N({\frak P}):=|D/{\frak P}|$. The prime ideal counting function of $D$ is given by $\pi_D(x)=\#\{{\frak P}\in{\rm ...

**3**

votes

**2**answers

182 views

### Linear polynomials in units of number fields

I would be thankful for a reference to any result that says "how often" an equation of the form $$c_1x_1 + c_2x_2 + ... + c_nx_n = 0,$$
where $n$ is fixed, $c_1, ..., c_n \in \mathcal{O}_K$ are ...

**11**

votes

**7**answers

804 views

### Properties of rings that have an elegant description in terms of the associated category of modules

Suppose $A$ is a ring. Then $A$ happens to be a division ring iff every left $A$-module is free. (See here for proofs). I think this is very beautiful; what other properties of rings have an elegant ...

**0**

votes

**1**answer

216 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 $m,n,$ ...

**1**

vote

**1**answer

105 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 ...

**1**

vote

**1**answer

134 views

### question about valuation ring

$k$ algebraically field, $A$ $k$ algebra and valuation ring of $K$ ($K$ field fraction of $A$) and we have the transcendence degree of $K$ over $k$ is one.
i want to ask if $A$ is noetherian ring?

**4**

votes

**2**answers

696 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 ...

**1**

vote

**1**answer

164 views

### Does the going-up theorem hold between flat algebras?

Let $R$ be a commutative Noetherian ring with unit and $S$ a flat $R$-algebra. Does the going-up theorem hold between $R$ and $S$?

**3**

votes

**2**answers

440 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 ...

**4**

votes

**2**answers

652 views

### The Weyl algebra modules which are also rings

Consider the space $C^\infty(\mathbb{R})$. We have on it a natural action of the Weyl algebra generated by $x$ and $d/dx$, where $x$ is the coordinate on $\mathbb{R}$, and there is plenty of Weyl ...

**0**

votes

**0**answers

113 views

### Cubic field and the corresponding cubic binary form

I am currently reading about binary cubic forms and cubic number fields (mainly about using binary cubic forms with integer coefficients to parametrize orders in the cubic field) and I thought it ...

**-4**

votes

**1**answer

104 views

### R is a commutative ring and every ideal is either R or 0. Show that R is a field [closed]

Assume that R is a commutative ring and every ideal is either R or 0. Show that R is a field.
How can I show this without first being given that $ 1\ne 0 \in R$

**0**

votes

**2**answers

262 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 ...

**3**

votes

**2**answers

425 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 ...

**1**

vote

**0**answers

90 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 ...

**1**

vote

**1**answer

153 views

### Do group identities of quotient with radical lift?

Let $R$ be a commutative Artinian ring and $J(R)$ its radical. Assume that the quotient $R/J(R)$ is a GI-ring.
(definitions that i use: I call a ring $S$ a GI-ring if its unit group, ...

**0**

votes

**1**answer

358 views

### Recursive Non-standard Models of Modular Arithmetic? [closed]

Any algebraically closed field (ACF) is a model of Modular arithmetic (MA). (MA) has the same axioms as first order Peano arithmetic (PA) except $\forall x(Sx \neq 0)$ is replaced with $\exists ...