In mathematics, in particular abstract algebra, a graded ring is a ring that is a direct sum of abelian groups $R_i$ such that $R_i R_j \subset R_{i+j}$. (Def: http://en.m.wikipedia.org/wiki/Graded_ring)

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### Cohomological criterion for non-triviality of negative part of graded module

Let $R$ be a graded ring and $M$ a graded module. Then for sufficently large $n$, we have $$H^0(\operatorname{Proj}(R), \widetilde{M}(n))\cong M_n.$$ Hence if I want to show that $M_{>0}$ is non-...
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### Reference for: a nontrivially graded integral domain is never quasi-local

Let $\Gamma$ be a torsionless grading monoid and $R=\bigoplus_{\alpha\in\Gamma}R_\alpha$ be a $\Gamma$-graded integral domain. I'm interested in the following result: if $R$ is nontrivially graded,...
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### Properties of graded rings and their (not necessarily graded) ideals (part 1)

I refer to Dummit Foote Chapter 7.4, Chapter 10.3 (pages 351,353,354,356 and 357) and Chapter 11.5 ($S$ is not necessarily unital or commutative). I also refer to my 2 preceding questions What is ...
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### Do we have $\mathbb ZA, RAR \subseteq$ Ideal$(A)$ subset of every ideal $J$ of $R$?

I refer to Dummit Foote Chapter 7.4, Chapter 10.3 (pages 351,353,354,356 and 357) and Chapter 11.5 ($S$ is not necessarily unital or commutative). Let $R$ be a ring, not necessarily commutative or ...
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### How to extend to a unique derivation on the graded tensor algebra $TV$?

If $V$ is a finitely generated graded module and $TV$ its graded tensor algebra then, Any degree $k$ linear map $V \to TV$ extends to a unique derivation of $TV$. I found this in page 45 of the book "...
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### What is internal direct sum or internal direct product in Dummit Foote?

I refer to Dummit Foote Chapter 10.3 specifically pages 351,353,354,356 and 357. Does Exercise 10.3.21 on pages 357 (By the way, there's some errata here. Condition (iii) should be $i_1,...,i_k$) ...
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### Motivation for Grading and Filtration

I have started reading about graded rings and modules and filtered rings and modules. For grading at least,I can see the polynomials as a prototype,graded by usual degree. But I can't seem to find ...
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### Associate graded module to a filtered module

I'm working through these notes on spectral sequences and I'm trying to make sure I understand the details regarding what the author calls the "associate graded module" $G_pM:=F_pM/F_{p-1}M$ ...
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### Can we control the number of homogeneous generators of a f.g. homogeneous ideal?

Let $G$ be an abelian group and $R$ be a $G$-graded ring. Question $1$: Is there a map $\phi:\mathbb{N}\rightarrow\mathbb{N}$ such that for every $n\in \mathbb{N}$ and any homogeneous ideal $I$ ...
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Let $f : S\to T$ be graded ring homomorphism of graded rings preserving degrees. Then is inverse image of every homogeneous prime ideal a homogeneous prime ideal?
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### Graded rings: what does $\mathbb{Z}[y]/(y, y^{2n + 1})$ with $y$ of degree $2$ mean?

I do not understand what $\mathbb{Z}[y]/(2y, y^{2n + 1})$ with $y$ of degree $2$ means. If I read the Wikipedia page right, the graded ring $\mathbb{Z}[y]$ is the set of all polynomials in $y$ with ...
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### invertible sheaves of the projective line

I'm trying to do the following exercise: "Consider the projective line $X=\mathbb{P}^1_R$ over a ring $R$. Describe Serre's twisted sheaves $\mathcal{O}_X(n)$, $n\in\mathbb{Z}$, via Cech cocycles and ...
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### Ring graded by a non-Abelian monoid

I'm looking for interesting examples of a $G$-graded ring where $G$ is a non-Abelian semigroup, monoid or group. Obvious examples are the semigroup algebra $kG$, but I haven't come across any others. ...
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### The associated graded ring of the localization $k[x_1,\dots,x_n]_{(x_1,\dots,x_n)}$

I was reading the Atiyah-Macdonald p. 121: Example. Let $A$ be polynomial ring $k[x_1,\dots,x_n]$ localized at the maximal ideal $\mathfrak{m}=(x_1,\dots,x_n)$. Then $G_{\mathfrak{m}}(A)$ is ...
Let $R$ be a commutative ring with a multiplicative identity. Let $A$ be a finitely generated graded $R$-algebra. Assume that $A_0$ is a finitely generated $R$-module. Is it true that $A_i$ is a ...