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This paper is the sequel to the paper “Commutative algebras of rational function matrices as endomorphisms of Kronecker modules I”. Here we focus on those Kronecker modules that are rank-2 extensions of finite-dimensional, torsion-free modules by the unique generic module R. We show that if such an extension is indecomposable, then its endomorphism algebra is trivial. This resolves the missing case from the previous paper, where the effects of height functions on the endomorphism algebras of extensions are studied.

In an abelian category with a torsion theory an object B is called a Baer object if Ext(B, T) = 0 for all torsion objects T. The Baer C[ξ]-modules are precisely the free C[ξ]-modules (P. Griffith). The modules over finite-dimensonal hereditary algebras of tame type have some properties analogous to C[ξ]-modules. However, in this paper it is shown that Baer modules over such algebras are not necessarily free or even pure-projective. A characterisation of Baer modules of countable rank over a particular Artin algebra is also given: A system of countable rank is Baer if and only if (V, W) = ∪n = 1∞ (Vn, Wn), where (Vn, Wn) is a finite-dimensional subsystem of (Vn + 1, Wn +1) and (Vn +1, Wn +1)(Vn, Wn) is torsion-free for n = 1, 2,…. Baer systems of arbitrary rank are also discussed.