Self-free modules and E-rings

We call an S -module M self-free with basis X if X is a subset of M and each function f : X → M extends to a unique endomorphism φ : M → M. Such structures were called minimally |X| -free in Bankston and Schutt (Bankston, P., Schutt, R. (1995). On minimally free algebras. Can. J. Math. 37(5):963-978.). If |X| = 1, then M is an E-module over S. We show that if M is slender, then M is a direct sum of |X| many copies of an E-module, without restrictions on the cardinality of X. We also investigate additive groups of torsion-free rings of finite rank without zero-divisors. We find criteria under which these rings are E-rings. Also, we find conditions for abelian groups to be E-groups.