$n$-Hopfian and $n$-co-Hopfian Abelian Groups

Abstract

For any natural number $n$ we define and study the two notions of $n$-Hopfian and $n$-co-Hopfian abelian groups. These groups form proper subclasses of the classes of Hopfian and co-Hopfian groups, respectively, and some of their exotic properties are established as well. We also consider and investigate $\omega$-Hopfian and $\omega$-co-Hopfian modules over the formal matrix ring.