Regular sequence of elements of degree 1 for a homogeneous Cohen-Macaulay ring

Assume that a positively graded ring R is generated in degree 1. Is it true that, if R is Cohen-Macaulay, then there exists a regular sequence x of elements of degree 1 so that R/x is zero dimensional?

I tend to believe that it holds, but could not find a reference. Maybe some extra condition should be imposed?

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You want to assume also that the residue field is infinite. Then a minimal reduction of the irrelevant ideal will be a system of parameters, each linear, and will be a maximal regular sequence.

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And over a finite field $k$, consider the union $V$ of all hyperplans of $\mathbb P^n_k$ and let $R$ be the obvious corresponding homogeneous algebra. Then $R$ is Cohen-Macaulay, but all elements of degree 1 are zero divisors. –  Qing Liu Oct 26 '10 at 12:27
Thanks to Graham and Qing! –  Li Li Oct 26 '10 at 13:39