  A pre-Lie-Rinehart algebra is an algebraic generalization of the notion of a
left-symmetric algebroid. We construct pre-Lie-Rinehart algebras from
r-matrices through Lie algebra actions. We study cohomologies of
pre-Lie-Rinehart algebras and show that abelian extensions of pre-Lie-Rinehart
algebras are classified by the second cohomology groups. We introduce the
notion of crossed modules for pre-Lie-Rinehart algebras and show that they are
classified by the third cohomology groups of pre-Lie-Rinehart algebras. At
last, we use (pre-)Lie-Rinehart 2-algebras to characterize the crossed modules
for (pre-)Lie Rinehart algebras.
