Given a commutative, $\mathbb N$-graded ring, one can associate to it a scheme via the $Proj$ construction. 


What happens if one tries to copy this procedure but instead of $\mathbb N$ with another indexing gadget (say commutative monoid) ?


Some thoughts about this:
Considering projective varieties is roughly the same as studying affine varieties equivariant under the multiplicative group. 
So I would guess that replacing $\mathbb N$ by something else corresponds to replacing the multiplicative group by something else.



Proj for rings graded by different things then $\mathbb N$ ?
Weighted projective spaces $\mathbb{P}(a\_1,\ldots,a\_n)$ are examples where a grading other than the standard grading is used. In general you can study gradings coming from any finitely generated abelian group, and this grading gives rise to a torus action on the ring. The Proj you speak of is then a GIT-quotient of $Spec R$ by this torus action (if you are familiar with GIT, the choice of linearization of the action corresponds to a choice of irrelevant ideal). This GIT-quotient construction is completely analogous to the usual construction
$$
\mbox{Proj} k[x\_0,\ldots,x\_n]=\left(\mbox{Spec} k[x\_0,\ldots,x\_n]-V(x\_0,\ldots,x\_n)\right) // \mathbb{G}\_m
$$
The best reference I can give for this stuff is Chapter 1 of the book ["Cox rings"](http://arxiv.org/abs/1003.4229) by Arzhantsev, Derenthal, Hausen and Antonio Laface. Other nice references are 


[The Homogeneous Coordinate Ring of a Toric Variety](http://arxiv.org/abs/alg-geom/9210008) by Cox, which deals with Toric varieties


[Lectures on invariant theory](http://books.google.co.uk/books?id=B9S_0iPvfrUC&printsec=frontcover&dq=dolgachev+invariant+theory&source=bl&ots=qxIPsqhMks&sig=59jcHL5Tn8AX6-N0H8EfixFSAco&hl=en&ei=NdLzTJCCMtGLhQeewJyxCA&sa=X&oi=book_result&ct=result&resnum=4&ved=0CCsQ6AEwAw#v=onepage&q&f=false)  by Dolgachev, which is a nice introduction to quotients in algebraic geometry.


