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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### Homogeneous elements of graded algebras as polynomials of algebra generators

Let $A = \oplus_{i > 0} A_{i}$ be a positively graded $k-$algebra and let $\mathcal{B} = \left\{b_{1}, \ldots, b_{n}\right\}$ be a generating set of $A$. Let $f$ be a homogeneous element in $A_{i}$ ...
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### How is $R_iR_j$ defined in the definition of graded rings?

In the definition of graded rings there is the condition $R_iR_j\subset R_{i+j}$, it involves the product of two subgroups $R_iR_j$. How is the set $R_iR_j$ defined? Edit: Is anyone aware of a ...
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### Prime avoidance for graded Noetherian ring with infinite residue field.

This question is related to this question. Let $R$ be a Noetherian local ring with infinite residue field. Let $\text{gr}_IR$ denote the associated graded ring $\bigoplus_{n=0}^{\infty} I^n/I^{n+1}$. ...
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### Finiteness of the homology of the Koszul complex

Let $A$ be a local noetherian commutative ring, let $x_1, \dots, x_r$ be elements of the maximal ideal of $A$, $I$ the ideal that they generate, $M$ a $A$-module of finite type such that $M/IM$ is of ...
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### Curious Difficulty with the Exterior Algebra (relationship between 2 and 3 forms)

Let $V$ be a finite dimensional vector space over $\mathbb{C}$. Then we can form the exterior algebra $\wedge V$. Let $L\subset \wedge^3 V$ be a linear subspace. I am working on the following claim. ...
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### I am a bit confused on the definition of a graded ring in Eisenbud.

He defines a graded ring as $R$ together with a direct sum decomposition $R = \bigoplus_{n \in \mathbb{N}}R_n$ where $R_n$ are abelian groups such that $R_iR_j \subset R_{i+j}$ I understand addition ...
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### Classification of graded fields, semifields, and skew fields

Recently I came upon the following result: Result 1. Let $K$ be a $\mathbb{Z}$-graded field. Then either $K$ is trivially graded (i.e. $K_k=0$ for $k\in\mathbb{Z}\setminus\{0\}$ with $K_0$ a field or ...
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### The associated graded of the ungraded polynomial ring is isomorphic to the ungraded polynomial ring as modules over the ungraded polynomial ring.

Let $k$ be a commutative ring. Let $k[t]$ be the ungraded polynomial ring with $\deg t=0$. The associated graded of the ungraded ring $k[t]$ is nevertheless graded and isomorphic to the graded ...
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I would like to discuss a fine observation that I made which is not discussed in the literarure. Maybe because it is easy. Nevertheless, I think that it is quite important. Let $k$ be a commutative ...
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### Maximal homogeneous ideals in graded ring A[T]

Let $A$ be a commutative algebra with unit. Let $A[x]$ be the polynomial algebra with coefficients in $A$ with the standard gradation (by degrees). I have the following questions. What are maximal ...
1 vote
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Let $A$ be a graded algebra over a commutative ring, and $I$ a two sided ideal of $A$ which is not necessarily graded. We can take the quotient $A/I$ and obtain an algebra, however I am struggling to ...
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### Symmetric Algebra of a Graded Module

Let $k$ be a commutative ring. Let $M$ be a $k$-module. Let $M$ be a graded module, concentrated in degree one. Then, the shifted module $M$ is concentrated in degree zero. Hence, $M$ is ...
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### Action on associated graded algebra inducing action on filtered algebra

Suppose $Q$ is a filtered algebra, with associated graded algebra $\text{gr}(Q)$. If we have an action of a ring $R$ on $\text{gr}(Q)$ (i.e. $\text{gr}(Q)$ is an $R$-module) then it seems clear that, ...
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### About the number of homogeneous generators of a f.g. homogeneous ideal?

Let $G$ be an abelian group and $R$ be a $G$-graded ring. We consider the following assertions. $(1)$ There is a map $\psi:\mathbb{N}\rightarrow\mathbb{N}$ such that for every $n\in\mathbb{N}$, and ...
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### Relation between definitions of a graded vector space and a graded module

I am new to the world of "grading", and my question is about the definition of the graded vector space. For definitions of these two, see graded vector space and graded module (they are ...
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### Does this map from a graded algebra have a name?

Let $k$ be a field and $A$ be a ($\mathbb{Z}$-)graded associative $k$-algebra. Suppose a $k$-linear map $\varphi : A \to k$ has the property that, whenever $a, b \in A$ are homogeneous of positive ...
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1 vote
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### Sufficient condition for radicality of an homogeneous ideal: still true when grading over arbitrary monoid?

Let $A$ be a graded commutative ring (if necessary, with unit). Suppose we have an homogeneous ideal $I\subset A$ that satisfies the property “for all homogeneous $f\in A$ such that $f^r\in I$ for ...
Let $A$ be a commutative ring, with unit, and suppose $A$ is graded over a commutative monoid $M$. (In particular $1\in A_e$, where $e\in M$ is the identity.) If $x\in A$ is invertible and homogeneous,...
### Questions about $K[x_1, \ldots , x_4]/(x_1x_2-x_3x_4)$ being a graded ring
So I was trying to understand this answer to the question of why $K[x_1, \ldots , x_4]/(x_1x_2-x_3x_4)$ ($K$ being a field) is not an UFD, and the author seems to use the fact that \$K[x_1, \ldots , ...