On rings whose faithful modules which are generated at most two elements are generators (Q2704586)

For a ring \(R\) consider the following property:NEWLINENEWLINENEWLINE(*) Every faithful right \(R\)-module which can be generated by at most two elements is a generator in the category of right \(R\)-modules.NEWLINENEWLINENEWLINEThe authors prove the following results:NEWLINENEWLINENEWLINE(1) Let \(R\) be a commutative Artinian ring with property (*). Then \(R\) is a quasi-Frobenius ring;NEWLINENEWLINENEWLINE(2) Let \(R\) be a commutative Noetherian ring with property (*). Then \(R=R_1 \times R_2\), where \(R_1\) is a finite direct product of Dedekind domains and \(R_2\) is a quasi-Frobenius ring.