# Tagged Questions

The commutative-rings tag has no usage guidance.

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### Let $f:R\to S$ be a local finite monomorphism .If $M$ is an Artinian $S$-module, is it an Artinian $R$-module?

$(R,m)$ and $(S,n)$ are local rings (commutative Noetherian with 1).
Let $f:R\to S$ be a local homomorphism/monomorphism ($f(m)=n$), such that the natural induced homomorphism $R/m\to S/n$ is an ...

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172 views

### GCD in polynomial vs. formal power series rings

I'm having problems finding an appropriate reference for this question.
Given two elements $f, g \in \mathbb{C}[x_1, \dots, x_n]$, consider their greatest common divisor, $\gcd_{\mathbb{C}[x_1, ...

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### How to prove prime avoidance for graded cases?

Let $R$ be a nonnegatively graded ring such that $R_0$ is local with infinite
residue field. Let $I,J_1,...,J_s$ be a homogeneous ideals of $R$ such that $I$ is not contained in $J_i$. Please prove ...

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100 views

### When a ring is a polynomial ring?

In the paper (2.11) the authors show that if $k^*$ is a separable algebraic extension of $k$ and $x_1,x_2, \ldots, x_n$ are indeterminates over $k^*$ and a normal one dimensional ring $A$ with $k ...

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### Result of Larsen and Lunts on rationality of power series with coefficients in a free abelian group

Let $G$ be a free abelian group (not necessarily finitely-generated) and $F$ be the fraction field of the group ring of $G$. Let $\Theta$ be the set of power series in $F[[t]]$ such that each nonzero ...

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### A paper by Y. Morita

The corresponding bibliographical details are:
Yoshihito Morita, Elementary proofs of the commutativity of rings satisfying $x^{n}=x$. Mem. Defense Acad. 18 (1978), no. 1, 1–24.
Does anybody here ...

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75 views

### Complexity of solving systems of linear diophantine equations

It is "well known" that a matrix system $Ax=b$ where $A\in \Bbb Z^{m\times n}$, $x\in \Bbb Z^n,b\in\Bbb Z^m$ for some $m,n \in \Bbb N$, can be solved in polynomial time, using Smith/Hermite Normal ...

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147 views

### Structure theorem for infinitely generated modules over a PID

This is a refined version of a question I asked days ago and have no answers yet. I am completely illietarte in algebra so, please, don't kill but explain.
The question is all in the title: is there ...

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51 views

### Decomposition of PID modules

This is (probably) the culmination of a series of questions I posted recently that have lead me to this (probably) final question. As usual, I aplogize for my illiteracy in algebra.
Recall that ...

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### When is the torsion submodule a direct factor?

Let $\mathbb{F}$ be a field (of characteristic 0, if needed) and $\mathbf{V}$ an $\mathbb{F}$-vector space. Let $T\in\mbox{End}_\mathbb{F}(\mathbf{V})$ be an endomorphism, and let (following Bourbaki) ...

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47 views

### Modification of nonfree locus

Let $ R $ be a commutative noetherian ring with identity. Let $ M $ be an $ R $-module. By definition the nonfree locus $ NF(M) $ of $ M $ is defined as the set of prime ideals $ {\mathfrak p} $ of $ ...

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68 views

### Cyclic decomposition of an infinitely generated module

My knowledge of algebra is undergraduate linear algebra, so I apologize for my complete ignorance.
Thinking about Jordan normal forms, I unintensionally came to an idea that turned out to be called a ...

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### Relative variants of the Jacobson radical

Let $B$ be a commutative ring (with 1). The Jacobson radical can be defined as
$$ J(B) = \{b \in B \mid \forall a \in B \colon \quad 1 + a\cdot b \text{ is a unit in } B \} $$ or $$ J(B) =\{ b \in B ...

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323 views

### invariants that can be measured by Local Cohomology

What invariants can be measured by Local Cohomology (and what application it has)?
As an example of what I mean:
Local Cohomology can measure invariants like depth and dim. So in some cases ...

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212 views

### A condition like primeness for zero ideal

Let $D$ be an integral domain (zero ideal is prime). Then for
every nonzero element $a,b \in D$, we have $\langle a\rangle\cap \langle b\rangle\neq 0$.
Now in a general case, let $R$ be a commutative ...

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185 views

### Maximal length of filter regular sequence

Let $A$ be a Noetherian ring and $R$ is a standard graded ring over $A.$ Let $M$ be a finitely generated graded $R$-module and $I$ be a graded ideal of $R.$ Then $x_1,\ldots,x_r\in I$ is called ...

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202 views

### On rings $R$ such that $xR\cap yR$ is non zero whenever $x$ and $y$ are non zero

I've isolated a property of rings (integral domains, associative, unitary, non necessarily commutative) that is useful to me :
$$xR\cap yR\neq\{0\}\quad\text{ whenever $x$ and $y$ are non zero.}$$
...

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45 views

### when: $depth\ R/I\ge depth\ R/J$ then $depth\ (R/I)_p\ge depth\ (R/J)_p$

Let $(R,m)$ be a Noetherian local ring, $M$ and $N$ finite R-modules, $p$ a prime ideal, and $I$ and $J$ ideals of $R$. Here, Count Dracula proves that in general assuming $depth\ R/I\ge depth\ ...

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186 views

### Rings such that all quotients by prime ideals are PIDs?

Let $R$ be a commutative ring such that for every prime ideal $P$ of $R$, the ring $R/P$ is a PID. Do you know how these rings are called or another characterization of them?
I know there are a lot ...

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791 views

### Has anything ever been done with the set $\{1,2,3,4,\ldots\}$ equipped with the operation $a \oplus b = a+b-1$ and the usual notion of multiplication?

Definition. $$\mathbb{J} = \{1,2,3,\ldots\}.$$
We can refer to the elements of $\mathbb{J}$ as "joiners."
The product of joiners is inherited from $\mathbb{Z}$.
The sum of joiners ...

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45 views

### Bass' stable range condition for principal ideal domains

Do you know a characterization of commutative rings $R$ whose every prime factor ring of $R$ is a principal ideal domain?

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107 views

### Completeness of Localizations of Completions of Commutative Rings

Let $R$ be an integral domain. Let $x,y\in R\setminus\{0\}$ be distinct. Let $\hat R$ be the $x$-adic completion of $R$ (the ring of all sequences $(r_n+Rx^n)_{n\ge0}$ where for $n\ge0$, $r_n\in R$ ...

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61 views

### A property of minimal prime ideals in rings with finite chromatic number

Let $R$ be a commutative ring with identity. There are so many ways to associate a graph to $R$. Consider this: take the elements of $R$ (All elements including zero) as vertices an two distinct ...

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54 views

### Analogue of Bass's Lemma 2.4 on when inverse images of free modules are free

Let $R$ be a Noetherian integral domain. Let $x\in R$ be a prime element. Let $\overline{R}=R/Rx$.
Let $P$ be a finitely-generated projective $R$-module.
Assume that $\frac{P}{xP}$ is a free ...

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111 views

### non-Noetherian r-Noetherian ring with Noetherian total quotient ring

A commutative ring is said to be r-Noetherian if every regular ideal is finitely generated, where an ideal is said to be regular if it contains a non-zerodivisor. Does there exist a non-Noetherian ...

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127 views

### What can be said about $A$ and $B$ given the exact sequence $0 \to R^p \to A \to R^r \to R^q \to B \to 0$?

Let $A,B$ be two $R$-modules over a commutative ring $R$ (restrict to $R = \mathbb{Z}$ or $R= \mathbb{K}$ a field where appropriate). Suppose $A$ and $B$ fit into an exact sequence
$0 \to R^p \to ...

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104 views

### When do all annihilators of primitive idempotents intersect in {0}?

maybe this is silly but:
for which class of rings (or commutative rings) R may I write
An element a of R is zero iff
for every primitive idempotent e, ea is zero
?
That is, primitive idempotents ...

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68 views

### Commutative noetherian domains with large fixed rings

Let $R$ be a commutative domain and let $\theta$ be a ring automorphism of $R$. The fixed ring of $\theta$ is defined by $R^{\theta}:=\{r \in R: \ \theta(r)=r \}.$ An ideal $I$ of $R$ is called ...

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357 views

### Is the sheaf of smooth functions flat?

Let $X$ be a smooth algebraic variety over $\mathbb{C}$. Is the sheaf of smooth functions on $X$ flat as an $\mathcal{O}_X$ module?

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### Grobner basis for a general algebra

Let $R$ be a quotient of the polynomial ring $\mathbb{C}[x_1,\dots , x_n]$. We fix a $\mathbb{C}^*$ action on $R$ which preserve homogenous components and the multiplication. (The geometric analogue ...

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146 views

### When does 'Zariski tangent space derivative' vanishes everywhere imply that a section is constant?

Consider an abelian algebra, $R$, over the field $K$ with the properties that every residue field of $R$ is (canonically) isomorphic to $K$ (I'm not sure but I think this is necessary, otherwise we ...

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234 views

### What kinds of limits does localization of commutative rings reflect?

Localization of commutative rings is a left exact left adjoint, so it behaves nicely with plenty of things. Local-to-global principles are also abundant in commutative algebra, and I thought some of ...

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### Reference request for $R$-index

Let $R$ be a noetherian domain with field of fractions $F$, let $V$ be a finite-dimensional $F$-vector space, and let $M,N \subseteq V$ be $R$-lattices in $V$ (finitely generated $R$-submodules of $V$ ...

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53 views

### Extending grading of subring to entire ring

Let $R$ be a (commutative) subring of $S$, and assume that $R$ is graded by an abelian group $G$. Is there anything known, possibly under less general circumstances, about the existence/uniqueness of ...

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175 views

### Can we Characterise Rings of Continuous Functions?

Suppose $K$ is some nice space, for example $\mathbb R$ or $\mathbb C$. Let $X$ be a set and $C$ a ring of functions $X \to K$. Is there any way to determine, from the algebraic structure of $C$, ...

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137 views

### A family of maximal ideals

Let $m_i $, $i \in I,$ be an infinite family of maximal ideals in a commutative ring with identity (it is not supposed to be Noetherian). When does there exist $j \in I$ such that $\cap_{i\not= j} ...

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89 views

### Theorem 16 , Chapter 5 of Northcott 's, Finite Free Resolutions: p.grade

Let $R$ be a commutative ring with identity. D.G. Northcott's, Finite Free Resolutions, has:
and in Theorem 16 of Chapter 5 proves that:
$p.grade(I,M) = p.grade(P,M)$ for some prime ideal $P$ ...

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679 views

### Examples of Noetherian overkill

I have read in many places that the noetherian hypothesis is often overkill - both in commutative algebra and in ($\overset?=$) algebraic geometry. In particular, I've read that coherence and finite ...

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### Generalization of a Result about degree bounds of invariant rings

A theorem of Knop states that if $G$ is semisimple and connected acting on a vector space $V$ over a field $K$ of characteristic 0, then the degree of the Hilbert series of $K[V]^G$ is less than or ...

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### Integer-matrix representation of a commutative ring

Consider a commutative ring $x_ix_j = N_{ij}^k x_k$, where $N_{ij}^k \in\{0,1,2,3,\cdots\}$, and $\{x_i\}$ is a finite set. (This is actually a fusion ring and $x_i$ are simple objects.)
How to find ...

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### Derivations annihilated by powers of the augmentation ideal

Consider an augmented commutative ring $R$, with augmentation ideal $\varpi$. Let $\delta$ be a derivation of $R$. The example I have in mind is $R=\mathbb F_p[x]/(x^{p^i})$ and $\delta=d/dx$, though ...

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110 views

### $Hom (T,R)$ isomorphic to $R:T$?

Let $R$ be a Cohen–Macaulay local ring with maximal ideal $m$ and $dim R = 1.$ In the paper "Almost Gorenstein rings", by "Goto, Matsuoka, Phuong", with this settings:
they have
How they reach the ...

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### A quick question from the paper “Generalizations of reductions and mixed multiplicities” by D. Rees

In the proof of Theorem 1.3 in the paper "Generalizations of reductions and mixed multiplicities" by Rees here, Is it necessary to consider the ring $Q'=Q/{\bigcup\limits_{q \geq 1}(0:{(I_1\dots ...

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### Upper bound for the minimum number of generators of the canonical module

Let $P=k[x_1...x_n]$ be a poly over a field. Suppose that $R=k[x_1...x_n]/I$. The canonical module of $R$ is $\omega_R=Ext^{n-dim(R)}_P(R,P)$.
The question is that is there any upper bound for the ...

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171 views

### In a noetherian commutative ring with only one associated prime, is the nilradical locally free?

The title says it all.
I suspect that the answer in general is no, although my intuition tells me that a jump in the dimension of the fibre of the nilradical at some point of Spec(A) can occur only ...

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### Torsion ideal in symmetric algebra

Let D be a a commutative domain, M be a D-module without torsion and S(M) its symmetric algebra. Is the D-torsion ideal of S(M) the prime ideal of S(M) ?

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123 views

### On factorization algorithms for $\mathcal{O}[x]$

We know that $\mathsf{LLL}$ algorithm provides factorization procedure that runs in poly time for polynomials in $\Bbb Z[x]$ that are primitive.
What other rings $\mathcal{O}$ can we use instead of ...

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106 views

### rings with 'flat functions'

Let $(R,\mathfrak{m})$ be a local ring over a field. Suppose the ring has flat elements, i.e. $\mathfrak{m}^\infty\neq\{0\}$. (The prototype is of course $C^\infty(\Bbb{R}^p,0)$, or a quotient of it, ...

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### Non-existence of commutative rings with many nilpotent elements with some restrictions where matrix powers are efficient

At the moment can't find better reference than "Cycle Enumeration using Nilpotent Adjacency Matrices with Algorithm Runtime Comparisons"
though certainly there are others.
Consider the following ...

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143 views

### Making idempotent element by a relation [closed]

Let $R$ be a commutative ring with identity and let $a, b \in R$ such that $a=ab$. How can we make a non zero idempotent element of $R$ by this relation?