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For a ring  R , the properties of being (left) self‐injective or being a cogenerator for the left  R ‐modules do not imply one another, and the two combined give rise to the important notion of pseudo‐Frobenius‐rings. For a coalgebra  C , (left) self‐projectivity implies that  C  is a generator for right comodules and the coalgebras with this property are 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  ∞  that generates its right comodules, and, if  C  is local over an algebraically closed field, then  C  ∞  can be chosen local as well. We also give some general conditions under which the implication ‘ C ‐projective (left) ⇒  C  generator for right comodules’ does work, and such conditions are when  C  is right semiperfect or when  C  has finite coradical filtration.

The generating condition for coalgebras