Covering coalgebras and dual non-singularity. (English) Zbl 1182.16026
We know that embedding algebras into better ones where certain problems have solutions is one of the major tools in ring theory. But an analogous tool for coalgebras does not always exist. Instead of embedding a coalgebra into a better behaved coalgebra one could try to find a suitable better behaved coalgebra with a projection – a covering coalgebra. Localisation is an important technique in ring theory and yields the construction of various rings of quotients. Several papers considered the colocalisation in comodule categories.
In this paper, the authors consider those possible coalgebra covers that could play the role of a coalgebra colocalisation. First, dual to the definition of a maximal dense extension of a module, the maximal codense cover is constructed for coalgebras with projective covers. Then, the authors consider a dual non-singularity concept for modules which turns out to be the comodule-theoretic property that turns the dual algebra of a coalgebra into a non-singular ring. At the end of this paper, coprime coalgebras and Hopf algebras which are non-singular as coalgebras are also discussed.
In this paper, the authors consider those possible coalgebra covers that could play the role of a coalgebra colocalisation. First, dual to the definition of a maximal dense extension of a module, the maximal codense cover is constructed for coalgebras with projective covers. Then, the authors consider a dual non-singularity concept for modules which turns out to be the comodule-theoretic property that turns the dual algebra of a coalgebra into a non-singular ring. At the end of this paper, coprime coalgebras and Hopf algebras which are non-singular as coalgebras are also discussed.
Reviewer: Zhang Liangyun (Nanjing)
MSC:
| 16T15 | Coalgebras and comodules; corings |
| 16S90 | Torsion theories; radicals on module categories (associative algebraic aspects) |
| 16D40 | Free, projective, and flat modules and ideals in associative algebras |
Keywords:
localisations of coalgebras; non-sigular coalgebras; hereditary coalgebras; path coalgebras; copolyform modules; maximal rings of quotientsReferences:
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