A left module M over a ring R
is cofaithful in case there is an embedding of R into a finite product of copies of M.
Our main result states that a semiprime ring R is left Goldie, that is, has a
semisimple Artinian left quotient ring, if and only if R satisfies (i) every faithful left
ideal is cofaithful and (ii) every nonzero left ideal contains a nonzero uniform left
ideal. The proof is elementary and does not make use of the Goldie and
Lesieur-Croisot theorems. We show that (i) and (ii) are Morita invariant. Moreover,
(ii) is invariant under polynomiaI extensions, and so is (i) for commutative rings.
Absolutely torsion-free rings are studied.
