# Split Lie-Rinehart algebras

## TL;DR

This paper introduces split Lie-Rinehart algebras, extending split Lie algebras, and demonstrates their decomposition into orthogonal sums of ideals, revealing their structure and relation to simple ideals.

## Contribution

It defines split Lie-Rinehart algebras and proves their decomposition into orthogonal sums of ideals, generalizing the structure theory of split Lie algebras.

## Key findings

- Decomposition of $L$ and $A$ into orthogonal sums of ideals
- Existence of a unique correspondence between ideals of $L$ and $A$
- Decomposition into simple ideals under mild conditions

## 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 $L = \bigoplus_{i \in I}L_i$, $A = \bigoplus_{j \in J}A_j$, 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 $A_{\tilde{i}}L_i \neq 0$. 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.