Coprime modules and comodules

Více o knize

The study of coalgebras arose from the theory of algebras and rings, aiming to transfer knowledge from algebras to coalgebras and from modules to comodules. However, the concept of primeness for rings and modules lacks a suitable counterpart in coalgebra due to the finiteness theorem for comodules, with only a few papers addressing this issue. By dualizing the classical primeness condition, coprimeness can be defined for modules and algebras, which we explore before applying the results to comodules and coalgebras over commutative rings. Notably, for any algebra A, being coprime as an A-module indicates that A is simple, whereas for a coalgebra C, the coprimeness condition as a comodule is less restrictive. We also examine the primeness of the endomorphism ring of an R-module M and its connection to M's primeness or coprimeness. For coalgebra C, the comodule endomorphism ring is isomorphic to the dual algebra C* with the convolution product, linking the primeness and coprimeness of both structures. We note that primeness conditions on comodules with non-zero socle and coprimeness conditions on those with proper radicals often yield trivial results. Additionally, we outline colocalization in module categories and apply it to comodules and coalgebras, emphasizing that in abelian categories, the existence of a colocalization functor relies on sufficient projectives. The role of a projective hull of a subgenerator in the dual case