Abstract
The concept of Hopfian modules has been extensively studied in the literature. In this paper, we introduce and study the notion of \(\delta \)-weakly Hopfian modules. The class of \(\delta \)-weakly Hopfian modules lies properly between the class of Hopfian modules and the class of weakly Hopfian modules. It is shown that over a ring R, every quasi-projective R-module is \(\delta \)-weakly Hopfian iff \(\delta (R)=J(R)\). We prove that any weak duo module, with zero radical is \(\delta \)-weakly Hopfian. Let M be a module such that M satisfies ascending chain condition on \(\delta \)-small submodules. Then it is shown that M is \(\delta \)-weakly Hopfian. Some other properties of \(\delta \)-weakly Hopfian modules are also obtained with examples.
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El Moussaouy, A., Hamzekolaee, A.R. ., Ziane, M. et al. Weak Hopfcity and singular modules. Ann Univ Ferrara 68, 69–78 (2022). https://doi.org/10.1007/s11565-021-00383-5
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DOI: https://doi.org/10.1007/s11565-021-00383-5