We give an interpretation of the Brauer group of a purely inseparable extension of exponent 1, in terms of restricted Lie-Rinehart cohomology. In particular, we define and study the category $p$-$\rm{ LR}(A)$ of restricted Lie-Rinehart algebras over a commutative algebra $A$. We define cotriple cohomology groups $H_{ p-LR}(L,M)$ for $L\in p$-$\rm{ LR}(A)$ and $M$ a Beck $L$-module. We classify restricted Lie-Rinehart extensions. Thus, we obtain a classification theorem for regular extensions considered by Hoshschild.
Categories: math.RA math.KT