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Let $R = \bigoplus_{n\in\mathbb{Z}}R_n$ be a graded ring. I'm trying to understand the structure of simple graded $R$-modules.

In C. Nastasescu and F. Van Oystaeyen book, Methods of graded rings, theorem 2.7.2 answer to this question : we can construct, from a simple $R_0$-module, a simple graded $R$-module. Moreover, every simple graded $R$-module arise from a simple $R_0$-module.

My question is the following one. If $M$ is a simple $R$-module (and not $R_0$-module), can we find a grading on $M$ such that $M$ is a simple graded $R$-module ?

Thanks for any help.

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    $\begingroup$ Not always. If $R=k[x]$, $k$ a field, then there are 1-dimensional $R$-modules $M$ where $x$ acts as multiplication by a constant ($xm=am$ for some $a \in k$). $\endgroup$
    – Dag Oskar Madsen
    Commented May 28, 2014 at 14:17
  • $\begingroup$ This is a simple $R_0$-module... but if $k$ admits non-squares, there are easy similar examples which are 2-dimensional over $k$. $\endgroup$
    – YCor
    Commented May 28, 2014 at 19:02

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