A ring \(R\) is called quasi-Frobenius if \(R\) is right (or left) Artinian and right (or left) self-injective. Define inductively the \(k\)th right socle \(S_ k\) of \(R\) as follows: \[ S_ 1=\text{Soc}(R_ R),\;S_ 2/S_ 1=\text{Soc}(R_ R/S_ 1),\;\dots,\;S_{k + 1}/S_ k=\text{Soc}(R_ R/S_ k)\;\dots \] It is well-known that a right perfect, right and left self-injective ring is quasi-Frobenius. In this paper, the authors replace the two-sided self-injectivity by either left self-injectivity or right self-injectivity, and prove the following result: Theorem. Let \(R\) be a right and left perfect right self-injective ring. Then \(R\) is quasi-Frobenius if and only if the second right socle \(S_ 2\) of \(R\) is finitely generated as a right ideal. Finally, the authors give an application of the theorem: Let \(R\) be a right and left perfect right self-injective ring. Then \(R\) is quasi-Frobenius if \(r(A)=r(J)\) for some finite subset \(A=\{a_ 1,\dots,a_ n\}\) of \(R\), where \(J\) is the Jacobson radical of \(R\).