The Generating Condition for Coalgebras

For a ring $R$, the properties of being (left) selfinjective or being cogenerator for the left $R$-modules do not imply one another, and the two combined give rise to the important notion of PF-rings. For a coalgebra $C$, (left) self-projectivity implies that $C$ is generator for right comodules and the coalgebras with this property were called right quasi-co-Frobenius; however, whether the converse implication is true is an open question. We provide an extensive study of this problem. We show that this implication does not hold, by giving a large class of examples of coalgebras having the "generating property". In fact, we show that any coalgebra $C$ can be embedded in a coalgebra $C_\infty$ that generates its right comodules, and if $C$ is local over an algebraically closed field, then $C_\infty$ can be chosen local as well. We also give some general conditions under which the implication "$C$-projective (left) $\Rightarrow C$ generator for right comodules" does work, and such conditions are when $C$ is right semiperfect or when $C$ has finite coradical filtration.