Vol. 44, No. 1, 1973

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Socle conditions for QF 1 rings

Claus M. Ringel

Vol. 44 (1973), No. 1, 309–336

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.

Mathematical Subject Classification
Primary: 16A36
Received: 24 September 1971
Published: 1 January 1973
Claus M. Ringel