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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### The degree $0$ component of a graded ring and localization

Let $S$ be a $\mathbb Z$-graded ring and $f \in S$ be a homogeneous element. Is the morphism $\mathrm{Spec}(S_{(f)}) \to \mathrm{Spec}(S_0)$ induced by the morphism of rings $S_0 \to S_{(f)}$ (which ...
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### Integers modulo prime power, not a graded ring?

We can decompose Z mod p^n as a vector space over Z mod p by writing numbers base p, eg, mod 3, 52= 27 + 2*9 + 2*3 + 1, and if Rm is {0,p^m,2p^m,....(p-1)p^m}, then we have for a in Rm, b in Rk, that ...
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### Interpretation of the numerator of the Hilbert series?

Let $R$ be a finitely generated graded ring over a field $k$. Let $R_\ell$ be the degree-$\ell$ homogeneous component of $R$. By the Noether normalization theorem, $R$ is finite over a graded ...
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Tom Marly here noted that if R is a graded field then $R$ is concentrated in degree 0, i.e., $R=R_0$ and $R_n =0$ for all $n \neq 0$. Is this proposition mentioned in any book or paper ?? ...
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### $a^m$ homogeneous implies $a$ homogeneous?

Let $A = \bigoplus_{r \in \mathbb{Z}/n\mathbb{Z}} A_r$ be an integral $\mathbb{Z}/n\mathbb{Z}$-graded ring of characteristic $0$, and $a \in A$. Assume there exists an integer $m$ such that $a^m$ is ...
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Let $A=\bigoplus_{n\geq 0}A_n$, $B=\bigoplus_{n\geq 0}B_n$ be two graded Noetherian rings, where $A_n\subset B_n$ and $A_0=B_0$ is a local ring. Suppose $B$ is a finitely generated $A$ module. Let $I$ ...
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Wikipedia define the graded ( ring, module, vector space, ...) as here I noted that in the rings and modules it required the condition of inclusion but in the vector spaces it did not. just the ...
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### (Reference) Does a ring's grading commute with the formation of fractions?

Question: Given a graded integral domain, when one forms the corresponding field of fractions, does the resulting field of fractions have a grading "compatible" with the original field of fractions?...
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### Given a proper homogeneous ideal, is there always a homogeneous maximal ideal that contains it? [duplicate]

It is known that every proper ideal of a ring must be contained in a maximal ideal, by the Zorn's lemma. Is this true in general for proper homogeneous ideals in a graded ring?
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### On giving a certain kind of grading to the polynomial ring over a commutative unital graded ring

For any commutative unital ring $R$ , $R[X]$ has a standard $\mathbb N \cup \{0\}$ grading where the $n$-th component is $R_n:= X^nR[X]$ ; however , given a commutative , unital , graded ring $R$ , I ...
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### When does commutativity in each homogenous component of a unital graded ring forces commutativity of whole ring?

Let $R=\oplus_{g \in G} R_g$ be a graded unital ring , graded by a monoid $G$ . Suppose $x_gy_g=y_gx_g , \forall x_g , y_g \in R_g ; \forall g \in G$ ; then is it true that $R$ is commutative ? If ...
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### Can a finitely generated graded module over commutative unital graded ring be also finitely generated by homogenous elements ?

Let $R$ be a commutative graded ring (https://en.wikipedia.org/wiki/Graded_ring) with unity , graded by a monoid (in particular $\mathbb Z$ or $\mathbb N \cup \{0\}$ if helpful for the purpose ) , ...
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I am looking for a proof of this fact: Let $W=k[x_1,\dots,x_n]$ be a positively graded polynomial ring. Every graded free resolution of a finitely generated $W$-module $M$ is isomorphic to the direct ...
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### Maschke's theorem for $G$-graded algebras

I am reading the paper Algebras graded by a group of Knus. Immediately I run into problems, which I will now detail: Let $G$ be a group and $K$ a field. A $G$-graded algebra $A$ is a finite-...
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Recently I read about graded rings and I read old papers but I noticed something all these papers define the graded ring but there is no proves (all rings are a group graded ring and satisfy the ...
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### Two-sided ideal $I$ in exterior algebra $T(V)/I$.

I have a confusion regarding two definitions of the two-sided ideal in exterior algebra. Def 1) In one definition, the exterior algebra $\Lambda(V)$ is defined as $T(V)/I$, where $I$ is the two-sided ...
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### Basic question on graded quotients

Let $A$ be a graded algebra, and let $I$ be a graded (two-sided) ideal. ($A=\bigoplus A_k$, $I=\bigoplus I_k$). Is it true that $A/I=\bigoplus A_k/I_k$? I can't seem to see it. Thanks for any help....
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### Correspondence of grading and group actions

Let $k$ be a field of characteristic zero with $n$-th roots of unity, $R$ a $k$-algebra and $G := \mathbb{Z}/n\mathbb{Z}$. To give an action of $G$ on $R$ is equivalent to grading $R$ by $G$. (To see ...
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### If $f \in \hat{m} =$ largest homogeneous ideal, then each coefficient in some generation is homogeneous.

Here is the notation & motivation.: Conjecture: If $f \in \hat{m}$, then $f = \sum g_i f_i$ where each $g_i$ is homogeneous. I've tried proof by induction on degree of $f$, number of ...
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### How is this proof correct in regard to a $k$-subalgebra (Eisenbud)?

This is testing my understanding of algebras, subalgebras, & polynomial rings. Page 31. Corollary 1.5. Let $k$ be a field, and let $S = k[x_1, \dots, x_r]$ be a polynomial graded by degree. ...
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### Basic fact used in proofs in Eisenbud's commutative algebra book. [duplicate]

Page 31. Corollary 1.5. Let $k$ be a field, and let $S = k[x_1, \dots, x_r]$ be a polynomial graded by degree. Let $R$ be a $k$-subalgebra of $S$. If $R$ is a summand of $S$, in the sense that ...
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I am currently reading Li Huishi's book "Zariskian filtrations". I am confused by a lemma (p. 34, Lemma 2 of section 4.3 in Chapter 1) in the book. It says: Let $S = \oplus_{n \in \mathbb{Z}}S_n$ ...