Characterization of finite Frobenius rings (Q5943638)

A ring \(R\) with identity is Frobenius if \(R\) is Artinian and satisfies both \(_R(R/J(R))\cong\text{Soc}(_RR)\) and \((R/J(R))_R\cong\text{Soc}(R_R)\), where \(J(R)\) denotes the Jacobson radical and \(\text{Soc}(-)\) denotes the socles. The author proves that a finite ring \(R\) is Frobenius if \(_R(R/J(R))\cong\text{Soc}(_RR)\). This generalizes the reviewer's recent Theorem 1 [in Indag. Math., New Ser. 9, No. 4, 627-628 (1998; Zbl 0919.16014)].