A ring R (associative with
identity) is called right ℵ-QF3 if it has a faithful right ideal which is a direct
sum of a family of injective envelopes of pairwise non-isomorphic simple
right R-modules. A right QF3 ring is just a right ℵ-QF3 ring where the
above family is finite. The aim of the present work is to give a structure
theorem for semiprime ℵ-QF3 rings. It is proved, among others, that the
following conditions are equivalent for a given ring R: (a) R is a semiprime
right ℵ-QF3 ring, (b) there is a ring Q, which is a direct product of right
full linear rings, such that SocQ ⊂ R ⊂ Q, (c) R is right nonsingular and
every non-singular right R-module is cogenerated by simple and projective
modules.
