Journal of the
Korean Mathematical Society JKMS

Hopf corings are defined in this work as coring objects in the category of algebras over a commutative ring $R$. Using the Dieudonn\'e equivalences from [7] and [19], one can associate coring structures built from the Hopf algebra $\mathbf F_p [x_0 , x_1, \ldots ]$, $p$ a prime, with Hopf ring structures with same underlying connected Hopf algebra. We have that $\mathbf F_p \![x_0 , x_1, \ldots ]$ coring structures classify thus Hopf ring structures for a given Hopf algebra. These methods are applied to define new ring products in the Hopf algebras underlying known Hopf rings that come from connective Morava $k$-theory.

Keywords: Hopf algebras, Hopf rings, Dieudonn\'e modules, homotopy theory

MSC numbers: 16W30, 57T05, 18E10