On the structure of graded $3$-Lie-Rinehart algebras

We study the structure of a graded $3$-Lie-Rinehart algebra $\mathcal{L}$ over an associative and commutative graded algebra $A\mathrm{.}$ For $G$ an abelian group, we show that if $(L,A)$ is a tight $G$-graded 3-Lie-Rinehart algebra, then $\mathcal{L}$ and $A$ decompose as $\mathcal{L}={⨁}_{i\in I}{\mathcal{L}}_i$ and $A={⨁}_{j\in J}{A}_j,$ where any ${\mathcal{L}}_i$ is a non-zero graded ideal of $\mathcal{L}$ satisfying $[{\mathcal{L}}_{i_1},{\mathcal{L}}_{i_2},{\mathcal{L}}_{i_3}]=0$ for any $i_1,i_2,i_3\in I$ different from each other, and any $A_j$ is a non-zero graded ideal of $A$ satisfying $A_j{A}_l=0$ for any $l,j\in J$ such that $j\mathrm{\ne }l,$ and both decompositions satisfy that for any $i\in I$ there exists a unique $j\in J$ such that $A_j{\mathcal{L}}_i\mathrm{\ne }0.$ Furthermore, any $({\mathcal{L}}_i,A_j)$ is a graded 3-Lie-Rinehart algebra. Also, under certain conditions, it is shown that the above decompositions of $\mathcal{L}$ and $A$ are by means of the family of their, respective, graded simple ideals.