NDLI: When Does the Rational Torsion Split Off for Finitely Generated Modules

Content Provider	Springer Nature Link
Author	Iovav, Miodrag Cristian
Copyright Year	2009
Abstract	It is well known that the torsion part of any finitely generated module over the formal power series ring K[[X]] is a direct summand. In fact, K[[X]] is an algebra dual to the divided power coalgebra over K and the torsion part of any K[[X]]-module actually identifies with the rational part of that module. More generally, for a certain general enough class of coalgebras—those having only finite dimensional subcomodules—we see that the above phenomenon is preserved: the set of torsion elements of any C -module is exactly the rational submodule. With this starting point in mind, given a coalgebra C we investigate when the rational submodule of any finitely generated left C -module is a direct summand. We prove various properties of coalgebras C having this splitting property. Just like in the K[[X]] case, we see that standard examples of coalgebras with this property are the chain coalgebras which are coalgebras whose lattice of left (or equivalently, right, two-sided) coideals form a chain. We give some representation theoretic characterizations of chain coalgebras, which turn out to make a left-right symmetric concept. In fact, in the main result of this paper we characterize the colocal coalgebras where this splitting property holds non-trivially (i.e. infinite dimensional coalgebras) as being exactly the chain coalgebras. This characterizes the cocommutative coalgebras of this kind. Furthermore, we give characterizations of chain coalgebras in particular cases and construct various and general classes of examples of coalgebras with this splitting property.
Ending Page	309
Page Count	23
Starting Page	287
File Format	PDF
ISSN	1386923X
e-ISSN	15729079
Journal	Algebras and Representation Theory
Issue Number	2-5
Volume Number	12
Language	English
Publisher	Springer Netherlands
Publisher Date	2009-03-26
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Local rings and generalizations Splitting Coalgebra Associative Rings and Algebras Torsion theories, radicals Torsion theory Non-associative Rings and Algebras Radicals and radical properties of rings Commutative Rings and Algebras Rational module Torsion theories; radicals on module categories
Content Type	Text
Resource Type	Article
Subject	Mathematics

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