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We study integral theory for left (or right) Hopf left bialgebroids. Contraryto Hopf algebroids, the latter ones don't necessary have an antipode $S$ but,for any element $u$, the elements $u_{(1)} \otimes S(u_{(2)})$ (or$u_{(2)}\otimes S^{-1}(u_{(1)})$ ) does exist. Our results extend those of G.B\"ohm who studied integral theory for Hopf algebroids. We make use of recentresults about left Hopf left bialgebroids. We apply our results to therestricted enveloping algebra of a restricted Lie Rinehart algebra.

Integral theory for left Hopf left bialgebroids