$\times_R$-Bialgebras associated with iterative $q$-difference rings

Realizing the possibility suggested by Hardouin [6], we show that her own Picard-Vessiot Theory for iterative $q$-difference rings is covered by the (consequently, more general) framework, settled by Amano and Masuoka [2], of artinian simple module algebras over a cocommutative pointed Hopf algebra. An essential point is to represent iterative $q$-difference modules over an iterative $q$-difference ring $R$, by modules over a certain cocommutative $\times_R$-bialgebra. Recall that the notion of $\times_R$-bialgebras was defined by Sweedler [17], as a generalization of bialgebras.