Hopficity and duo rings (Q2243062)

In this paper, the authors study the notion of Hopfian modules for right duo rings and right chain rings. They prove that a right chain ring $R$ is a right duo ring if and only if every cyclic right $R$-module is Hopfian. They also prove that if $R$ is either a right chain ring or a right duo ring then $R_R$ is Hopfian. If $R$ is a ring with finite right Goldie-dimension they prove that every finitely projective right $R$-module as well as every finite dimensional, nonsingular right $R$-module is Hopfian. From this result they deduce that if $R$ is either a local ring or a ring with finite right Goldie-dimension then a free right $R$-module is Hopfian if and only if it is finitely generated. They generalize this result to right duo and right chain ring as follows. If $R$ is a right duo and right chain ring then for any index set $J$, a right $R$-module of the form $\bigoplus_{i\in J}x_iR$ is Hopfian if and only if $J$ is finite. The authors conclude the paper with the following result: If $R$ is a local, right duo, right Artinian, right FGC-ring then a right $R$-module $M$ is Hopfian if and only if $M$ is finitely generated (here a ring $R$ is right FGC-ring if every right $R$-module is a direct sum of cyclic modules).