Vol. 30, No. 3, 1969

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Two characterizations of quasi-Frobenius rings

Edgar Andrews Rutter

Vol. 30 (1969), No. 3, 777–784

The purpose of this paper is to characterize quasi-Frobenius, QF, rings in terms of relationships assumed to exist for each cyclic or finitely generated left module between the module and its second dual, where duality is with respect to the ring. More specifically we prove that a left perfect ring is QF if every cyclic left module is reflexive or every finitely generated left module is (isomorphic to) a submodule of a free module. For rings with minimum condition on left or right ideals this later condition is equivalent to every finitely generated left module being torsionless or to the ring being a cogenerator in the category of finitely generated left modules. If annihilator relations are defined by means of the natural pairing between a module and its dual, this condition is also equivalent to every submodule of a finitely generated left module being an annulet.

Mathematical Subject Classification
Primary: 16.55
Received: 26 March 1968
Published: 1 September 1969
Edgar Andrews Rutter