Split Lie-Rinehart algebras

Abstract

We introduce the class of split Lie-Rinehart algebras as the natural extension of the one of split Lie algebras. We show that if $L$ is a tight split Lie-Rinehart algebra over an associative and commutative algebra $A,$ then $L$ and $A$ decompose as the orthogonal direct sums $L={⨁}_{i\in I}{L}_i$, $A={⨁}_{j\in J}{A}_j$, where any $L_i$ is a nonzero ideal of $L$, any $A_j$ is a nonzero ideal of $A$, and both decompositions satisfy that for any $i\in I$ there exists a unique $\tilde{i}\in J$ such that $A_{\tilde{i}}{L}_i\ne 0$. Furthermore any $L_i$ is a split Lie-Rinehart algebra over $A_{\tilde{i}}$. Also, under mild conditions, it is shown that the above decompositions of $L$ and $A$ are by means of the family of their, respective, simple ideals.