Generalizations of 𝑄𝐹-3 algebras

Colby, R. R.; Rutter, E. A.

doi:10.1090/S0002-9947-1971-0269686-5

Generalizations of $\textrm {QF}-3$ algebras
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by R. R. Colby and E. A. Rutter

Trans. Amer. Math. Soc. 153 (1971), 371-386

DOI: https://doi.org/10.1090/S0002-9947-1971-0269686-5

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Abstract:

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 $\Sigma$-injective left ideal, $R$ contains a faithful $II$-projective injective left ideal; the injective hulls of projective left $R$-modules are projective, and the projective covers of injective left $R$-modules are injective. Moreover, these rings are shown to be semi-primary 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.

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