Proj for rings graded by different things then $\mathbb N$ ? 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.
 A: First let's revisit the usual case. The $\mathbb N$-grading on $R$ says that
$Spec\ R$ is a cone, and there is a map $R \to R_0$. We can rip out 
$Spec\ R_0$ from $Spec\ R$, take the quotient, and get a reasonable space
$Proj\ R$.
If we grade instead by say $({\mathbb N})^k$, then there are $k$ analogous
quotients of $R$, and $k$ closed subschemes to rip out before we divide
by $({\mathbb G}_m)^k$. This comes up if e.g. $R = Fun(G/N)$, with the
torus acting on the right (through the maximal unipotent group $N$),
and the corresponding quotient is $G/B$.
Some of the things I would look
up here are "Cox coordinate ring of a toric variety" and "Mori dream space",
but I don't know the references well enough to suggest a best place.
A: 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" by Arzhantsev, Derenthal, Hausen and Antonio Laface. Other nice references are 
The Homogeneous Coordinate Ring of a Toric Variety by Cox, which deals with Toric varieties
Lectures on invariant theory  by Dolgachev, which is a nice introduction to quotients in algebraic geometry.
