Large E-rings exist (Q1821205)

A ring R with 1 is called an E-ring provided \(Hom_{{\mathbb{Z}}}(R,R)=R\) under the map \(\phi\to 1\phi\). E-rings were defined by \textit{P. Schultz} [J. Aust. Math. Soc. 15, 60-69 (1973; Zbl 0257.20037)]. They are of interest for the ''theory'' of torsion-free abelian groups, mainly in case of groups of finite rank. This connection was observed in earlier work by \textit{R. S. Pierce} [Mich. Math. J. 7, 241-243 (1960; Zbl 0103.268)]. Obvious examples of E-rings are subrings of the rationals. The only classical examples are quite similar, which restricts their size drastically \((\leq 2^{\aleph_ 0})\). In this paper E-rings are constructed for all infinite cardinals \(\lambda^{\aleph_ 0}\). The construction is based on the notion cotorsion-free and a combinatorial method, ''Shelah's Black Box'' which replaces additional set theoretic assumptions as the ''Jensen functions'' [see \textit{A. L. S. Corner}, \textit{R. Göbel}, Proc. Lond. Math. Soc., III. Ser. 50, 447-479 (1985; Zbl 0562.20030)]. The interesting construction is similar to recent constructions of indecomposable abelian groups.