Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Generalizations of $ {\rm QF}-3$ algebras


Authors: R. R. Colby and E. A. Rutter
Journal: Trans. Amer. Math. Soc. 153 (1971), 371-386
MSC: Primary 16.40
MathSciNet review: 0269686
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 16.40

Retrieve articles in all journals with MSC: 16.40


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1971-0269686-5
PII: S 0002-9947(1971)0269686-5
Keywords: QF-$ 3$ ring, left perfect ring, unique minimal faithful module, faithful $ \Sigma $-injective left ideal, faithful $ II$-projective injective left ideal, injective hull projective, projective cover injective duality, projective module, endomorphism ring, hereditary ring, semihereditary ring
Article copyright: © Copyright 1971 American Mathematical Society