Two-sided E-rings

Let R be a ring, 1 R, and R+ the additive group of R. We define Mult(R) to be the subring of End(R+) generated by all left and right multiplications by elements of R. The ring R is called a two-sided E-ring if End(R+)=Mult(R). If R is torsion-free of finite rank (tffr), we call R a quasi-two-sided E-ring if . We investigate (quasi)-two-sided E-rings, give several examples and construct large two-sided E-rings R with prescribed center S such that End(R+)=Mult(R)≈R SRop≈R. Thus our rings R are examples of two-sided E-rings that are weak E-rings as well, i.e. R≈ End(R+), but R is not an E-ring.