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Rings whose faithful left ideals are cofaithful

1975; Mathematical Sciences Publishers; Volume: 58; Issue: 1 Linguagem: Inglês

10.2140/pjm.1975.58.1

1945-5844

John A. Beachy, W.D. Blair,

Algebraic structures and combinatorial models

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 polynomial extensions, and so is (i) for commutative rings.Absolutely torsionfree rings are studied.