Prime elements in partially ordered groupoids applied to modules and Hopf algebra actions

The notion of a prime element in a ring is a semigroup notion as its definition only involves the multiplication of the ring. Also the concept of a prime ideal can be expressed in terms of the multiplication of ideals using the fact that the lattice of ideals of a ring is partially ordered. Associativity does not play a role here and hence the concept of prime elements in partially ordered groupoids (p.o.-groupoids) occurs naturally. Here a groupoid means a set with a binary operation (= non-associative semigroup). Bican, Jambor, Kepka and Nemec defined in 1980 a binary operation on the lattice of submodules of a module that turns this lattice into a p.o.-groupoid. Thus prime submodules of a module can be defined in terms of primeness of elements of a p.o.-groupoid. We will shortly report on modules whose lattice of submodules is a prime p.o.-groupoid. Semiprime groupoids are defined analogous and we compare those concept when transferred to modules with other "semiprime" notions for modules. Eventually we will apply our concepts to the study of Hopf module algebras, i.e. algebras that admit an action of a Hopf algebra. In particular we will explain a Theorem of Bergman and Isaac on group actions in module theoretic terms. Moreover we will point out the relationship of the obtained results and an open question on the semiprimeness of smash products.