2005/12/02 by Tomasz Brzezinski, Brzezinski, Tomasz, Ryan B. Turner +1
Mathematics · #16W30 #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #math.QA #math.RA #msc:16W30
paper · pdf · doi:10.48550/arxiv.math/0512049
27 pages, LaTeX
arxiv created 2005/12/02 · openalex publication_date 2005/12/02 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A theory of monoids in the category of bicomodules of a coalgebra C or C-rings is developed. This can be viewed as a dual version of the coring theory. The notion of a matrix ring context consisting of two bicomodules and two maps is introduced and the corresponding example of a C-ring (termed a \em matrix C-ring) is constructed. It is shown that a matrix ring context can be associated to any bicomodule which is a one-sided quasi-finite injector. Based on this, the notion of a \em Galois module is introduced and the structure theorem, generalising Schneider's Theorem II [H.-J. Schneider, Israel J. Math., 72 (1990), 167--195], is proven. This is then applied to the C-ring associated to a weak entwining structure and a structure theorem for a weak A-Galois coextension is derived. The theory of matrix ring contexts for a firm coalgebra (or \em infinite matrix ring contexts) is outlined. A Galois connection associated to a matrix C-ring is constructed.