Subgroups of the Baer-Specker Group with Prescribed Endomorphism Ring and Large Dual

This paper generalises a result obtained in a recent paper [ 6 ]. Given a countable ring A with free abelian additive group, we here construct a pure subgroup G of the Baer-Specker group ℤ ^ω which gives a split realisation of A, in the sense that End G =A⊕ Fin G where Fin G denotes the ideal of all endomorphisms of finite rank, subject to the further conditions that |End G| = 2^ℵ₀ while the dual G* = Hom(G,ℤ) is as large as it could be, viz. of cardinal 2^ℵ₀ . The construction relies on a result of independent interest, namely that a countably generated algebra over a PID R with free underlying R-module is always isomorphic to an algebra of row-and-column-finite matrices over R.