A Review To Some Properties of Semiperfect and Quasi- Frobenius Rings

Abstract

In [15] Ikeda showed that a ring R is left and right mininjective and left and right Artinian if and only if it is quasi-Frobenius. In [33] Utumi proved that any left and right continuous and left and right Artinian ring is quasi-Frobenius. In [7], Camillo and Yousif generalized Ikeda’s and Utumi’s result as a left and right continuous ring with ascending chain condition on left annihilators is quasi- Frobenius ring. It is natural to ask whether Faith’s or Camillo and Yousif ’s results can be extended to semiperfect CS-rings. However, it is a result of G´omez Pardo and Guil Asensio [29] that every right Kasch, right CS-ring has a finitely generated essential right socle, but it is unknown weather a right Kasch right CS-ring is always semiperfect [31]. In this paper, we review and prove some properties on Kasch and quasi-Frobenius rings which is related to these results.