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Let R be an associative ring
with 1. If RM is a left R-module, then M can be considered as a right 𝒞-module,
where 𝒞 = Hom(RM,RM) is the centralizer of RM. There is a canonic ring
homomorphism ρ from R into the double centralizer 𝒟 = Hom(M𝒞,M𝒞) of RM. For
a faithful module RM, the homomorphism ρ is injective, and RM is called balanced
(or to satisfy the double centralizer condition) if ρ is surjective. An artinian
ring R is called a QF-1 ring if every finitely generated faithful R-module is
balanced. This definition was introduced by R. M. Thrall as a generalization of
quasi-Frobenius rings, and he asked for an internal characterization of QF-1
rings.
The paper establishes three properties of QF-1 rings which involve the left
socle and the right socle of the ring; in particular, it is shown that QF-1
rings are very similar to QF-3 rings. The socle conditions are necessary and
sufficient for a (finite dimensional) algebra with radical square zero to be QF-1,
and thus give an internal characterization of such QF-1 algebras. Also, as a
consequence of the socle conditions, D. R. Floyd’s conjecture concerning the
number of indecomposable finitely generated faithful modules over a QF-1
algebra is verified. In fact, a QF-1 algebra has at most one indecomposable
finitely generated faithful module, and, in this case, is a quasi-Frobenius
algebra.
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