Let R be a ring with
identity. R is left QF-3 if R has a minimal faithful (left) module, i.e., a faithful (left)
module, which is (isomorphic to) a summand of every faithful (left) module. We
show that left QF-3 rings are characterized by the existence of a faithful
projective-injective left ideal with an essential socle which is a finite sum of simple
modules. The main result is a structure theorem for left and right QF-3 rings with
zero left singular ideal. This theorem gives several descriptions of this class of rings.
Among these is that the above rings are exactly the orders (containing units) with
essential left and right socles in semi-simple two-sided (complete) quotient
rings.