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For a ring $R$ and a graded ideal $I$, let $I_{\geqq p}= \oplus_{i\geqq p}I_i$.

If $R/I$ is an artinian, is $R/I_{\geqq p}$ artinian?

If it is false, then is it true in polynomial ring case?

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Yes, in the case of polynomial rings, what you ask is true.

More precisely, if $R=k[X_1,...,X_n]$ is a polynomial ring over a field and $I\subset R$ is a graded ideal, then $R/I \;\text {artinian}\implies R/I_{\geqq p} \;\text {artinian}$.

Indeed, $R/I$ artinian means that the only prime ideals containing $I$ are maximal ideals . (Geometrically this says that the subscheme $V(I)\subset \mathbb A^n_k$ has dimension zero)
Now we have $ I_{\geqq p}\supset I^p $ so that any maximal ideal containing $I_{\geqq p}$ must contain $I$ and is thus maximal.
This proves that the ring $R/I_{\geqq p}$ is artinian too.

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I have given a counterexample to the general question in another, independent answer. – Georges Elencwajg Aug 2 '12 at 12:01

No, $R/I_{\geqq p}$ needn't be artinian.

Let $k$ be a field and define $$R=k[X_1,X_2,...,X_n,...]/\langle X_iX_j\mid i,j\geq 1\rangle=k[x_1,x_2,..., x_n,...]$$ This quotient of the polynomial ring in infinitely many indeterminates inherits a graded structure from the polynomial ring (with $\text {deg} \:x_i=1$) since we have factored out a homogeneous ideal.

Now if $I=\langle x_i\mid i\geq 1\rangle$, we have $R/I=k$ which is certainly artinian.
However $I_{\geqq 2}=0$ so that $R/I_{\geqq 2}=R$, which is not artinian (nor even noetherian).

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thanks Georges, but I saw at Irena peeva "Graded syzygies" p.80 in this book it is true. but I don't reason. – Sang Cheol Lee Aug 2 '12 at 9:40
Dear Sang, I am afraid I don't quite understand what you mean. What "is true" in that reference by Irena peeva ? And what do you mean by "but I don't reason" ? – Georges Elencwajg Aug 2 '12 at 9:46
In Irena peeva book, it is true. but I don't know reason which is artinian. – Sang Cheol Lee Aug 2 '12 at 10:09
I have addressed the case of polynomial rings in another, independent answer. – Georges Elencwajg Aug 2 '12 at 10:20

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