A ring R with 1 is called an E-ring provided Hom,(R, R) z R under the map cp + 1~. The class of E-rings was defined and studied by Schultz [S] in 1973, and further investigated by Bowshell and Schultz [BS] in 1977. Examples of E-rings include Z/(n), subrings of Q, and pure subrings of the ring of p-adic integers. More interesting examples are the torsion-free E-rings of finite rank. These were characterized in [BS] as those rings quasi-isomorphic to R, x R, x . . . x R,, where each Ri is a strongly indecomposable subring of an algebraic number field and Hom,(R,, Rj) = 0 for i #j. In spite of their seemingly specialized nature, such rings have played an important role in the theory of torsion-free groups of finite rank, dating back to Beaumont-Pierce [BPl, BP2], and Pierce [P] in 1960-1961. (See also [APRVW, NR, PV, RI.) Relatively little has been published on infinite rank torsion-free E-rings. Indeed, until recently, the only examples of these were provided by the pure subrings of the p-adic integers. In this paper, the “Black Box” of Shelah is used to construct a host of large E-rings. We show that any torsion-free p-reduced, p-cotorsion-free, commutative ring S may be embed88 0021-8693/87 $3.00