Let's say we have a graded ring $R=\bigoplus_{i=0}^n R_i$ ($R_m=0$ for $m > n$), in particular we have $R_i R_j \subset R_{i+j}$.
We now have a $R$-module $M$ which is also graded $M=\bigoplus_{i=0}^m M_i$, but instead of the common condition $R_iM_j \subset M_{j+i}$ we have a somehow "inverted" condition $R_iM_j \subset M_{j-i}$ for $i \geq j$ (or for all $i,j$ by setting $M_k=0$ for $k<0$).
I tried to somehow rearrange or invert $R$ to get the original definition, but it didn't work out: We could inverse the grading of $R$ to get $R'=\bigoplus_{i=-n}^0R_{-i}$ which would give a real graded $R'$-module $M$ with $R'_iM_j=R_{-i}M_j\subset M_{j+i}$. However, $R'$ is no more a graded ring because only non-negative indices are allowed for the grading. How can this dilemma be solved?
Is $M$ with this negativity condition still a graded $R$-module? If yes, how can we prove that and if no, are there any counter examples or is there a special name for this type of "modules"? I would appreciate any help!