# Tagged Questions

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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Can someone give me a construction of the coproduct of two (not necessarily commutative) GRADED algebras $A,B$ over a commutative ring $k$? I haven't found it on the internet. Thank you!
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### Hopf gradings on complex commutative group rings

Let $G$ be a finite abelian group. The complex group ring $\mathbb{C}G$ admits a structure of Hopf algebra when the multiplication is the usual multiplication in a group ring and the co-multiplication ...
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### $\Bbb Z$-graded ring with no nonzero homogeneous prime ideals

Exercise $2.18$ in Eisenbud's algebra book asks to prove: Suppose $R=\bigoplus_{n=-\infty}^\infty R_n$ is a $\Bbb Z$-graded ring such that any homogeneous prime ideal is zero. Prove $R_0$ is a field. ...
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### Well-definedness of homogeneous coordinate ring of projective scheme [closed]

Let $I,J \subset S=k[x_0,...,x_n]$ be homogeneous ideals. How can I show that $\operatorname{Proj}S/\bar{I}=\operatorname{Proj}S/\bar{J}$ iff $\bar{I}=\bar{J}$? (Here $\operatorname{Proj}R$ is the ...
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### Homogeneous localization and regularity

Let $k$ be a field, $S = k[x_0,\dots,x_r]$, $I$ a homogeneous ideal of $S$ and $R=S/I$. Let $P$ be a homogeneous prime ideal of $R$ and let $R_{(P)}$ be the homogeneous localization of $R$ at $P$. I ...
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### $0 \in S_k$ for which k?

If $S$ is a graded ring, for which $k \in \mathbb{Z}$ do we have $0 \in S_k$? I think there shouldn't exist such a k. So as 0 is the empty sum we don't need this?
Let $M$ and $N$ be two modules (can assume them to be finitely generated if need be) over the ring $A=k[x_0,...,x_n]$. Denote by $E(M)$ the injective hull of $M$. We work in the category of positively ...