Vol. 58, No. 1, 1975

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Rings whose faithful left ideals are cofaithful

John Allen Beachy and William David Blair

Vol. 58 (1975), No. 1, 1–13

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.

Mathematical Subject Classification
Primary: 16A08
Received: 22 February 1974
Published: 1 May 1975
John Allen Beachy
William David Blair