# Questions tagged [graded-rings]

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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### Graded field over integers

In a $\mathbb Z$-graded field $R$, prove that $R=R_0$. Use the fact that all units in a graded domain are homogenous or otherwise. Try: Let $x_n$ be any nonzero element. Then it is homogeneous and ...
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### How to regard $\Bbb{C}[x,y,z]/(xy-z^2)$ as graded ring? [duplicate]

I heard $\Bbb{C}[x,y,z]/((xy-z^2)$ can be regarded as grade ring, but how ? $\Bbb{C}[x,y,z]$ can be regarded as graded ring by decomposition$\Bbb{C}[x,y,z]＝ \bigoplus_{d\geq 0} A_d$, where $A_d＝${...
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### Necessary and Sufficient condition for Flatness of Graded Modules

I am currently self studying the textbook Commutative Algebra with a View Toward Algebraic Geometry by David Eisenbud and came across the following exercise: Exercise 6.10 (Flatness of graded modules):...
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### Why is $S_{0}$ a subring of the graded ring $S=\bigoplus_{d\geq 0}S_{d}$. [duplicate]

An (essentially) the same question has been asked and answered here Definition of graded rings, but the user who posted the answer has not been here for a while, and I am confused by the answer, so I ...
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### Question about lengths in graded rings

Let $A$ be a graded, noetherian ring and $\mathfrak p$ a minimal (minimal in the set of all prime ideals) homogeneous ideal. Is it true that the rings $A_{\mathfrak p}$ and $A_{(\mathfrak p)}$ have ...
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### Global dimension of a polynomial ring, knowing its graded global dimension

Let $R$ be an $\mathbb{N}$-graded noetherian ring. In what follows, "global dimension" means the supremum of the projective dimensions of $R$-modules (left or right; it doesn't matter), ...
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### Let $\mathfrak p, \mathfrak p' ⊂ S$ be relevant prime ideals. If $\mathfrak p \cap S_+ = \mathfrak p' \cap S_+$, then $\mathfrak p = \mathfrak p'$.

Let $S$ be a graded ring. Let $\mathfrak p, \mathfrak p' ⊂ S$ be relevant prime ideals. If $\mathfrak p \cap S_+ = \mathfrak p' \cap S_+$, then $\mathfrak p = \mathfrak p'$. I don't see how this is ...
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### Let $S$ be a $\mathbb Z^{\ge 0}$-graded ring over $A$. If $S_+$ is finitely generated, then $S$ is a finitely generated graded $A$-algebra?

Let $S$ be a $\mathbb Z^{\ge 0}$-graded ring over $A$. Suppose $S_+= (a_1, \dots, a_n)$ where $a_i$ are homogeneous of positive degree. I want to show that $S$ is a finitely generated graded $A$-...
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### Ideals in localisation of graded ring

Let $k$ be a field and $I$ a homogeneous ideal of $k[x_1,...,x_n]$. Consider the graded ring $R=k[x_1,...,x_n]/I$ with unique homogeneous maximal ideal $\mathfrak m=(x_1,...,x_n)R.$ Let $J$ be a ...
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### $\Omega^*(M)$ as a graded algebra over the ring of smooth functions. Is there a notion of basis?

In Introduction to Smooth Manifolds by John M. Lee p.360, he defines $\Omega^*(M) =\displaystyle\bigoplus_{k=0}^n\Omega^k(M)$, the differential forms on a manifold $M$, as being an associative, ...
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### If $x^n \in R$ for some $n \in \mathbb{N}$, do all of the homogeneous components $x_i$ of $x$ also satisfy $x_i^n \in R$?

Let $R \subseteq S$ be an extension of $\mathbb{N}$-graded rings. Let $x = x_0 + x_1 + \dots + x_m$ ($x_i \in S_i$) satisfy $x^n \in R$ for some $n \in \mathbb{N}$. Does it follow that $x_i^n \in R$ ...
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Let $R:=\mathbb{C}[x_1,\ldots,x_d]$ be a polynomial ring. Let $I=(f_1,\ldots,f_n)$ be an $R$-ideal. Let $f^*_i$ be the homogenization of $f_i$ in $R[t]$. Question: If $(\overline{f_1^*},\ldots,\... • 1,368 2 votes 1 answer 76 views ### Example of a graded ring with respect to any completely additive arithmetic function - is it isomorphic to anything? Let$\varphi : \Bbb{N} \to \Bbb{N}$be a completely additive arithmetic function, i.e.$\varphi(nm) = \varphi(n) + \varphi(m)$for all$n, m \in \Bbb{N}$. For example,$\Omega(n) = $the total number ... • 19.3k 0 votes 1 answer 98 views ### Algebraically closed proof. Let$B = \bigoplus_{i \in \mathbb Z}B_i$be a$\mathbb Z$-graded integral domain. Given$f \in B$define$\deg: B \rightarrow \mathbb Z \cup \{-\infty\}$by$$\deg(f) = \mathrm{maxSupp}{(f)} \text{ if ... 0 votes 1 answer 220 views ### proving$\deg(fg) = \deg(f) + \deg(g)$in a$\mathbb Z$-graded integral domain. In a$\mathbb Z$-graded integral domain$B = \bigoplus_{i \in \mathbb Z} B_i$my definition for the degree function is as follows: Given$f \in B,$define$\deg : B \rightarrow \mathbb Z \cup \{-\... 1 vote
### Proving that $\deg(f) = - \infty$ iff $f=0.$
In a $\mathbb Z$-graded integral domain $B = \bigoplus_{i \in \mathbb Z} B_i$ my definition for the degree function is as follows: Given $f \in B,$ define \$\deg : B \rightarrow \mathbb Z \cup \{-\... 