I've been reading about $\mathbb{Z}$-graded rings and modules recently and have attempted to understand these as sequences of abelian groups with connecting maps $A_i\times M_j\to M_{i+j}$ etc. This approach seems to be rather cumbersome the way I'm doing it compared to regarding them as actual rings and modules, so I might have got the wrong idea.
Maybe I'm looking in the wrong places but everywhere I look they are defined simply via direct sum decompositions, and I can't find any systematic treatment of this topic using the 'sequence of abelian groups' approach. Where can I read about this?
Here is how I have defined a graded-commutative ring (I am working in characteristic two for the moment): For each $i\in\mathbb{Z}$, an abelian group $A_i$ and for each $i,j$, a biadditive map $\pi_{ij}:A_i\times A_j\to A_{i+j}$ for which $$\pi_{i+j,k}(\pi_{ij}(a,b),c)=\pi_{i,j+k}(a,\pi_{jk}(b,c)),$$ $$\pi_{0i}(1,a)=a,$$ $$\pi_{ij}(a,b)=\pi_{ji}(b,a).$$ (Note that $\pi_{00}$ is a ring multiplication on $A_0$, so the $1$ here is based on the additional requirement that $A_0$ is unital.)
The paper "On the Structure of Hopf Algebras" by Milner and Moore (Ann. Math. 1965) takes this approach. See Section 1.