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 $A.$ 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 $\mathcal{L} =\bigoplus_{i\in I}\mathcal{L}_i$ and $A =\bigoplus_{j\in J}A_j,$ 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 $j\neq l,$ and both decompositions satisfy that for any $i\in I$ there exists a unique $j\in J$ such that $A_j \mathcal{L}_i\neq 0.$ 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.