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I'm looking for a reference in commutative algebra for the properties of the ring made of polynomials in $n$ indeterminate over a field $k$ with "real exponents".

I don't even know the name of this ring, and I would like to know which properties hold for it and for the modules over it.

Edit : As explained by YCor, it's the (semi)group algebra $k[\mathbf{R}^n_+]$ over the field $k$.

I would also like to add that my modules and ring are graded, exactly in the same way that $k[x, y]$ is a graded ring.

Thanks,

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  • $\begingroup$ Can you please define this ring? $\endgroup$
    – Laurent Moret-Bailly
    Commented Jun 30, 2016 at 15:06
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    $\begingroup$ It's the group algebra over $k$ of the additive group of $\mathbf{R}^n$, sometimes denotes $k[\mathbf{R}^n]$, or, if you only allow nonnegative exponents, it's the semigroup algebra $k[\mathbf{R}_+^n]$ over $k$ of the additive $\mathbf{R}_+^n$, where $\mathbf{R}_+=[0,+\infty[$. As a $k$-algebra, it has no memory of the structure of the reals, $\mathbf{R}^n$ being isomorphic to $\mathbf{Q}^{(c)}$ additively ($c$ = continuum), it's isomorphic as $k$-algebra to $k[\mathbf{Q}^{(c)}]$, which is a bit ugly. $\endgroup$
    – YCor
    Commented Jun 30, 2016 at 15:17
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    $\begingroup$ (I removed the redundant tag "commutative-rings"; for those with enough (I don't know how many) reputation points please upvote the synonym suggestion mathoverflow.net/tags/synonyms?filter=suggested&tab=newest so that commutative-rings will automatically be changed to ac.commutative-algebra) $\endgroup$
    – YCor
    Commented Jun 30, 2016 at 22:07

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