# Tagged Questions

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### Polynomial ring operations on $\mathbb{Z}$

I have asked this on Math Stack Exchange but without answers: The usual ring operations on $\mathbb{Z}$ can be defined via polynomials in $\mathbb{Z}[a,b]$ (when viewing $a,b$ as variables): ...
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### 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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### 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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### 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 $M$-...
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### 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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### 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 r-...
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### 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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### 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 ...
### $Hom (T,R)$ 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 ...