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Embedding torsionless modules in projectives

Publicacions Matemàtiques
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Publicacions Matemátiques, Vol 34 (1990), 379-387 . Abstract EMBEDDING TORSIONLESS MODULES IN PROJECTIVES CARI, FAITH In this paper we study a condition right FGTF on a ring R, namely when all finitely generated torsionless right R-modules embed in a free module . We show that for a von Neuman regular (VNR) ring R the condition is equivalent to every matrix ring R� is a Baer ring ; and this is right-left symmetric . Furthermore, for any Utumi VNR, this can be strengthened : R is FGTF iff R is self-injective . Introduction When all injective modules over a ring R embed in a free R-module, the ring R must be quasi-frobenius, (= QF) and conversely . In this case every right or left R-module embeds in a projective R-module, and furthermore, the one-sided condition implies the two-sided condition ([F-W]) . The two-sided condition that all cyclic R-modules embed in projectives implies R is QF, but here the one-sided condition is not sufficient . Related to these rings are right IF rings initiated by Jain [J], Colby [Col, Damiano [D], and Würfel [W] . A ring R is right IF if every (injective) right module embeds in a fiat module, Le . if every injective right R-module is fiat . These rings were characterized by Colby and Würfel in the one-sided case by the property that all finitely presented right R-modules embed in projective modules . Here, again, right does not imply the left condition ([Co]) . However, two-sided IF are coherent (I .c .) . A number of characterizations of IF rings are summarized in Section 4 . In this paper we study a condition that is considerably weaker, namely right FGTF: all finitely generated torsionless right R-modules embed in a projective equivalently in a free right R-module . We let right FPTF denote the same condition for finitely presented torsionless right R-modules . We now introduce a concept that is germane to FGTF: A ring R is a right *-ring (star ring) provided that every finitely generated right R-module has finitely generated dual . Easy exe

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