On Coprime Modules and Comodules

Many observations about coalgebras were inspired by comparable situations for algebras. Despite the prominent role of prime algebras, the theory of a corresponding notion for coalgebras was not well understood so far. Coalgebras C over fields may be called coprime provided the dual algebra C∗ is prime. This definition, however, is not intrinsic it strongly depends on the base ring being a field. The purpose of the paper is to provide a better understanding of related notions for coalgebras over commutative rings by employing traditional methods from (co-)module theory, in particular (pre-)torsion theory. Dualising classical primeness condition, coprimeness can be defined for modules and algebras. These notions are developed for modules and then applied to comodules. We consider prime and coprime, fully prime and fully coprime, strongly prime and strongly coprime modules and comodules. In particular we obtain various characterisations of prime and coprime coalgebras over rings and fields.