QcF-coalgebras. (Q1884523)

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.