Weak Hopfcity and singular modules (Q2133616)

The authors, in this paper, generalizing the notions of weakly Hopfian module and $\delta$-Hopfian module, introduce the notion of $\delta$-weakly Hopfian module and study its properties. They derive some equivalent conditions for a projective module to be $\delta$-weakly Hopfian. They also prove that every (quasi-)projective $R$-module is $\delta$-weakly Hopfian if and only if $R$ has no nonzero semisimple projective $R$-module (which is equivalent to the condition that $\delta(R)=J(R)$). The authors prove that $\delta$-weakly Hopficity is not closed under taking factor modules. However, they prove the following theorem. If $M$ is a quasi-projective, uniform $R$-module and $N$ is a nontrivial, fully invariant, small submodule of $M$ then the factor module $M/N$ is Hopfian and is hence $\delta$-weakly Hopfian. In the statement of this theorem the hypothesis that ``$M$ is uniform'' is missing though it is included in the statement of this theorem given in the Introduction. The authors conclude the paper by generalizing for $\delta$-weakly Hopfian modules the well known result that every Noetherian module in Hopfian. In some of the results proved by the authors stronger assertion ``Hopfian'' holds. I don't understand why the authors have included ``singular modules'' in the title of the paper as they won't study in detail ``singular modules'' in this paper. Actually ``singular module'' occurs in the definition of a $\delta$-small submodule and only in Example 2.22 singular module appears.