A Note on Coseparable Coalgebras
Jawad Abuhlail

TL;DR
This paper explores the relationship between coseparable coalgebras and corings, demonstrating that a coalgebra can be viewed as a coring over its dual and characterizing coseparability via the existence of counits.
Contribution
It establishes a novel connection between coseparable coalgebras and corings, providing conditions for the existence of counits in the associated coring.
Findings
A coalgebra over a commutative ring can be viewed as a coring over its dual.
Coseparability of a coalgebra is equivalent to the coring having a counit.
The paper characterizes when the coring has a left or right counit.
Abstract
Given a coalgebra over a commutative ring we show that can be considered as a (not necessarily counital) -coring. Moreover, we show that this coring has a left (right) counity if and only if is coseparable as an -coalgebra.
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Taxonomy
TopicsRings, Modules, and Algebras · Algebraic structures and combinatorial models · semigroups and automata theory
