Assume $R$ is a graded ring and the $M_i$ are graded modules. Then Bruns and Herzog define the graded direct product $^*\Pi M_i$ as the submodule of $\Pi M_i$ generated by the sequences $(x_i)$ with $x_i$ homogeneous.
I can't understand the difference between the graded direct product and the normal product, could you give me an example when they are different?
Take $R=M_i=k[x]$. Define $p_i=1+x+\cdots x^i$. I think that $(p_i)_i$ is an element of the product but not an element of the graded product (because you can't have infinite sums).