• DOI:10.2307/1995563
• Corpus ID: 53518461

GENERALIZATIONS OF QF-3 ALGEBRAS

@inproceedings{Colby2010GENERALIZATIONSOQ,
  title={GENERALIZATIONS OF QF-3 ALGEBRAS},
  author={Robert R. Colby and Edgar A. Rutter},
  year={2010}
}

• R. Colby, E. A. Rutter
• Published 2010
• Mathematics

This paper consists of three parts. The first is devoted to investigating the equivalence and left-right symmetry of several conditions known to characterize finite dimensional algebras which have a unique minimal faithful representation— QF-3 algebras—in the class of left perfect rings. It is shown that the following conditions are equivalent and imply their right-hand analog: R contains a faithful S-injective left ideal, R contains a faithful LT-projective injective left ideal; the injective hulls of projective left Ä-modules are projective, and the projective covers of injective left .R-modules are injective. Moreover, these rings are shown to be semiprimary and to include all left perfect rings with faithful injective left and right ideals. The second section is concerned with the endomorphism ring of a projective module over a hereditary or semihereditary ring. More specifically we consider the question of when such an endomorphism ring is hereditary or semihereditary. In the third section we establish the equivalence of a number of conditions similar to those considered in the first section for the class of hereditary rings and obtain a structure theorem for this class of hereditary rings. The rings considered are shown to be isomorphic to finite direct sums of complete blocked triangular matrix rings each over a division ring. Thrall [36] called an algebra A of finite rank (left) QF-3 if it has a unique minimal faithful left module, that is, a unique (up to isomorphism) module M with the property that no proper direct summand is faithful. It is not difficult to verify the equivalence of the following statements. (1) A is (left) QF-3. (2) A contains a faithful injective left ideal. (3) The injective hull of AA is projective. (4) The injective hull of every projective left ^4-module is projective. (5) The projective cover of every injective left ^-module is injective. Moreover, the right-hand analogue of each of these conditions is easily established by forming duals with respect to the field so that for algebras being QF-3 is a two-sided property and distinctions between left and right are unnecessary. It is apparent from the above list that there are several possible ring theoretic generalizations of QF-3 algebras and this paper is primarily devoted to resolving certain questions which arise naturally in connection with such extensions. Presented to the Society, January 24, 1970 under the title Endomorphism rings of projective modules; received by the editors April 11, 1969 and, in revised form, November 25, 1969. AMS 1969 subject classifications. Primary 1610, 1650.