Vol. 28, No. 2, 1969

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QF-3 rings with zero singular ideal

Robert Ray Colby and Edgar Andrews Rutter

Vol. 28 (1969), No. 2, 303–308

Let R be a ring with identity. R is left QF-3 if R has a minimal faithful (left) module, i.e., a faithful (left) module, which is (isomorphic to) a summand of every faithful (left) module. We show that left QF-3 rings are characterized by the existence of a faithful projective-injective left ideal with an essential socle which is a finite sum of simple modules. The main result is a structure theorem for left and right QF-3 rings with zero left singular ideal. This theorem gives several descriptions of this class of rings. Among these is that the above rings are exactly the orders (containing units) with essential left and right socles in semi-simple two-sided (complete) quotient rings.

Mathematical Subject Classification
Primary: 16.53
Received: 15 April 1968
Published: 1 February 1969
Robert Ray Colby
Edgar Andrews Rutter