Picard-Vessiot extensions of artinian simple module algebras

This paper pursues Takeuchi's Hopf algebraic approach [M. Takeuchi, A Hopf algebraic approach to the Picard-Vessiot theory, J. Algebra 122 (1989) 481-509] to the Picard-Vessiot (PV) theory for differential equations, to involve the PV extensions of difference equations. Differential fields and total difference rings in the standard PV theory are unified here by artinian simple (AS) module algebras over a cocommutative, pointed smooth Hopf algebra.