On Coprime Modules and Comodules

Abstract

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 article 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.

Dualizing 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.

Key Words:

• (Co)Prime modules and comodules
• Coprime coalgebras
• Dual algebras

2000 Mathematics Subject Classification:

• 16W30
• 16D60

ACKNOWLEDGMENTS

Parts of the presented material are taken from the first author's Ph.D. thesis written at the Heinrich Heine University of Düsseldorf (2006). She wants to express here sincere thanks to the German Academic Exchange Service (DAAD), to Gahjah Mada University, and to the Freunde und Förderer der HHU for financial support and to the Mathematical Institute of HHU for warm hospitality.

Notes

Communicated by T. Albu.