Recently I read about graded rings and I read old papers but I noticed something all these papers define the graded ring but there is no proves (all rings are a group graded ring and satisfy the condition $R_g.R_h \subseteq R_g+R_h$). So,
- Is there any ring that's not a $G$-graded ring?
- If the answer is no, how can I proof that the inclusion condition is satisfied for any ring and group $G$ (I mean here the non-trivial graded ring)?
Maybe my question would be trivial, but I need a clear vision.
We claim that $\Bbb Z$ is not a $G$-graded ring for any nontrivial monoid $G$.
Suppose it is; then in particular $\Bbb Z=\bigoplus A_g$ for some subgroups $A_g$ of $\Bbb Z$ with $A_1\neq\Bbb Z$. But then the injection $A_1\to\Bbb Z$ is given by multiplication by some number $m$, which implies that $A_*=\bigoplus_{g\neq 1} A_g$ has a nonzero torsion element (since, e.g. $0\to A_0\to\Bbb Z\to A_*\to 0$ is a split exact sequence). But since $\Bbb Z$ has no torsion, we have a contradiction.
The proof goes through for semigroups, of course, since there was nothing essential used about the identity element here.