$\newcommand\ZZ{\mathbb{Z}}$Let $R=\bigoplus_{n\in\mathbb N_0}R_n$ be your graded ring. Construct a category $Q$ with objects $\ZZ$ and where $\hom_Q(n,m)=R_{m-n}$ with composition coming from the multiplication in $R$. A graded left $R$-module is the same thing as a functor from $Q$ to abelian groups. The (non-unital) ring associated to $Q$ does what you want, and if you prefer you can add a unit to it.
$\newcommand\ZZ{\mathbb{Z}}$Let $R=\bigoplus_{n\in\mathbb N_0}R_n$ be your graded ring. Construct a category $Q$ with objects $\ZZ$ and where $\hom_Q(n,m)=R_{m-n}$ with composition coming from the multiplication in $R$. A graded left $R$-module is the same thing as a functor from $Q$ to vector spacesabelian groups. The (non-unital) algebra ring associated to $Q$ does what you want, and if you prefer you can add a unit to it.
$\newcommand\ZZ{\mathbb{Z}}$Let $R=\bigoplus_{n\in\mathbb N_0}R_n$ be your graded ring. Construct a category $Q$ with objects $\ZZ$ and where $\hom_Q(n,m)=R_{m-n}$ with composition coming from the multiplication in $R$. A graded left $R$-module is the same thing as a functor from $Q$ to vector spaces. The (non-unital) algebra associated to $Q$ does what you want, and if you prefer you can add a unit to it.