Abstract
We study the structure of a graded 3-Lie-Rinehart algebraLover 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 L and A decompose as L = ? i?I Li and A = ? j?J Aj, where any Li is a non-zero graded ideal of L satisfying [Li1 ,Li2 ,Li3] = 0 for any i1, i2, i3 ? I different from each other, and any Aj is a non-zero graded ideal of A satisfying AjAl = 0 for any l, j ? J such that j ?l, and both decompositions satisfy that for any i ? I there exists a unique j ? J such that AjLi ? 0. Furthermore, any (Li,Aj) is a graded 3-Lie-Rinehart algebra. Also, under certain conditions, it is shown that the above decompositions of L and A are by means of the family of their, respectively, graded simple ideals.
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