Coprime modules and comodules

The study of coalgebras was motivated by the existing theory of algebras and rings, and by transferring the corresponding knowledge from algebras to coalgebras and from modules to comodules. However, the notion of primeness for rings and modules did not find an adequate counterpart in the coalgebraic setting due to the finiteness theorem for comodules. Only few papers were dealing with this problem. Dualizing to the classical primeness condition, coprimeness can be defined for modules and algebras. We investigate the resulting notions for modules and then transfer the outcome to comodules and coalgebras over commutative rings. Notice that for any algebra A, coprimeness as an A-module implies that A is a simple algebra, but for a coalgebra C the condition to be coprime as a comodule is not so restrictive. Also of interest is the primeness of the endomorphism ring of an R-module M and its relationship with the primeness or coprimeness of M. In the case of a coalgebra C, the comodule endomorphism ring of C is isomorphic to the dual algebra C* with the convolution product. In this situation the question reduces to the interplay of primeness and coprimeness conditions of the coalgebra C and the dual algebra C*. We observe that primeness conditions on comodules with non-zero socle and coprimeness conditions on comodules with proper radicals lead to trivial situations. Studies of localization and colocalization of coalgebras over a field were done by some authors with respect to coidempotent subcoalgebras of C. To avoid the dependence on the base ring being a field, we give an outline of colocalization in module categories and then apply it to comodules and coalgebras. In abelian categories the existence of a colocalization functor depends on the presence of enough projectives in the category. We transfer the technique of colocalization in the category of R-modules to the comodule situation. The question arises about the role of a projective hull of a subgenerator in the dual case. However no comparable constructions are possible in this situation. In fact, the existence of a projective hull of a strongly coprime coalgebra implies the trivial result.