On Rings Whose Proper Factors Are Semisimple

A ring R is called left PFS if left proper factors of R are semisimple. We will observe that being PFS is not a left-right symmetric property. Moreover, we observe that a left PFS ring is either a semisimple ring or a ring with only one non-zero left ideal or a domain. We show that R is a left PFS domain if and only if every R-module is a direct sum of an injective and a torsion free modules. When R is a domain, R is a left PFS ring if and only if R is a left SI-ring if and only if R is a left V-ring with K-dimR=1 if and only if R is a left PCI ring. In case R has only one left ideal, it is a strictly left weakly V-ring, and also R is a strictly left and right weakly V-ring if and only if R is a left and right PSF ring. Among other things, we prove that a ring R is a Camillo-Krause left weakly V-ring if and only if R is a left PFS ring. This provides a partial answer to Camillo-Krause problem.