On the structure of split regular Hom-Lie Rinehart algebras

The aim of this paper is to study the structures of split regular Hom-Lie Rinehart algebras. Let $(L,A)$ be a split regular Hom-Lie Rinehart algebra. We first show that $L$ is of the form $L=U+{\sum }_{[\gamma ]\in \mathrm{\Gamma }\mathrm{/}\sim }{I}_{[\gamma ]}$ with $U$ a vector space complement in $H$ and $I_{[\gamma ]}$ are well described ideals of $L$ satisfying $I_{[\gamma ]},I_{[\delta ]}=0$ if $I_{[\gamma ]}\mathrm{\ne }{I}_{[\delta ]}$. Also, we discuss the weight spaces and decompositions of $A$ and present the relation between the decompositions of $L$ and $A$. Finally, we consider the structures of tight split regular Hom-Lie Rinehart algebras.