A graded ring R is the direct sum of additive subgroups Rd, as d runs from 0 to infinity. Each d extracts a slice of R. An element in Rd is called homogeneous of degree d. Thus R is the direct sum of homogeneous elements. A typical entry might be x+y+z, where x is in R3, y is in R5, and z is in R9.
Each Rd is closed under addition, and the product of two homogeneous elements adds their degree, or yields 0. The product rule tells us 1 lives in R0, along with all the integers up to the characteristic of R. In fact R0 is a subring of R. Note that each Rd is an R0 module.
At first this looks like a valuation ring, where the product attains the sum of the individual valuations, but there is a difference. If x and y have the same valuation, the valuation of x+y could be larger. This cannot happen to homogeneous elements in a graded ring. Localize Z about P, and give it the usual valuation. Now 1 has valuation 0, but if you add 1 to itself P times, you get something with the valuation 1. Conversely, the polynomials in x form a graded ring (see below), yet this is not a local ring, and not a valuation ring.
The typical example is the ring of polynomials in one or more indeterminants over some other ring K. A polynomial lives in Rd if every term has degree d. Thus x2+xy+yz lives in R2. Verify that every polynomial is a unique direct sum of these homogeneous polynomials. The subring R0 is equal to K.
Let c be a real number strictly between 0 and 1, and let the distance between x and y be cn, for the greatest n such that Hn contains x-y. This is determined by the homogeneous component of x-y with lowest degree.
Since 0 is in every Rd, let 0 be homogeneous with infinite degree. Hence its norm is c∞, or 0. The distance from x to y is 0 iff x = y. (Remember that each nonzero x-y lives in some Hn, according to the lowest degree term in x-y.)
Consider the triangular inequality on three points x y and z. Write z-x as z-y + y-x. The valuation of the sum, on the left, is at least as large as the lesser valuation on the right. The distance on the left is no larger than the smaller distance on the right. Therefore we have a metric space, with its open ball topology.
Given a cauchy sequence S, the component from Rd settles down to a constant value. This holds for every d, hence S defines a canonical series of elements from Rd for all d. The partial sums of this infinite series defines another cauchy sequence T. Verify that S-T converges to 0. In other words, T is the same as S. Treat the completion of R as the set of formal power series in a dummy variable w, where the coefficient on wd comes from Rd.
If you want addition and multiplication to be continuous, and backward compatible with the ring operations of R, then two power series in w are added in the usual way (per component), and multiplied in the usual way. The result is a continuous ring. I'm glossing over a lot of δ ε details here, but I went through them once in valuation rings, and I don't feel like doing it again. It's pretty intuitive in any case.
When R is the polynomials over K, its completion is the formal power series, in the same indeterminants, with coefficients in K.
A graded R module M is based on a graded ring R, as described above. Write M as the direct sum over Md. Once again, each Md is closed under addition. Furthermore, the degree of cx is the degree of c (in R) plus the degree of x (in M), or cx = 0.
Let Hn be the submodule that is the direct sum of Md, as d runs from n to infinity. Build a metric on M, just as we did on R. This turns M into a metric space. The completion of M becomes formal power series in w, with coefficients in Md. These series are not multiplied together, but they can be multiplied by a polynomial that represents an element of R. When a coefficient in Rd is multiplied by a coefficient in Md, invoke the action of R on M. In other words, the action is determined by the action of each homogeneous element of R on each homogeneous element of M.
A graded ring R is a graded R module; simply let R act on itself.
A homomorphism between graded modules takes homogeneous elements of degree d (outside the kernel) to homogeneous elements of degree d+d0, for some offset d0, specific to the homomorphism. Thus the graded R modules form a subcategory of R modules.
The descending chain of ideals in R, or submodules in M, cannot stabilize at anything other than 0. A nonzero x in every Hn is the sum of homogeneous parts, and at some point, the lowest homogeneous component drops away, and x disappears, leading to a smaller ideal. Thus R, or M, is artinian iff its nth ideal or module becomes 0.
If R is an integral domain, let x have nonzero degree. The powers of x are nonzero, the chain of descending ideals cannot stabilize at 0, and R is not artinian.
The product of homogeneous elements is homogeneous by definition. The converse holds in an integral domain. If xy is homogeneous, look at the lowest and highest degree terms of x times y, and conclude x and y are homogeneous.