QcF-coalgebras.
type: Item
instance of: scholarly article
title: QcF-coalgebras.
zbMATH Open document ID: 1064.16043
published in: Southeast Asian Bulletin of Mathematics
publication date: 2004-11-01T00:00:00Z
review text: Any coalgebra \(C\) over a field \(k\) is a bimodule over its dual algebra \(C^*=\Hom_k(C,k)\). The coalgebra \(C\) is said to be left Quasi-co-Frobenius (QcF for short) if there exists a monomorphism of left \(C^*\)-modules from \(C\) into a free left \(C^*\)-module. This notion was introduced and investigated by \textit{C. Năstăsescu} and the reviewer [in J. Algebra 174, No. 3, 909-923 (1995; Zbl 0833.16038)], and the present paper adds some characterizations of QcF coalgebras to those given there. The most relevant asserts that left QcF coalgebras are those left semiperfect coalgebras (i.e. its category of left comodules has enough projectives) for which every projective left comodule is injective and no left comodule has injective dimension equal to \(1\) (Theorem 2.1). Co-semisimple coalgebras are characterized as left hereditary left QcF coalgebras (Proposition 2.4). The uniqueness of integrals for QcF coalgebras is discussed in Section 3. Some other consequences of well-known results are deduced in Section 4. -- There are several inaccuracies in the references.
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16L60
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 2113351
zbMATH Keywords: quasi-co-Frobenius coalgebras
zbMATH Keywords: semiperfect coalgebras
zbMATH Keywords: integrals
author: Mingyi Wang
reviewed by: José Gómez-Torrecillas
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2113351