ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 32, Number 4, Winter 2002 M -FREE ABELIAN GROUPS MANFRED DUGAS, SHALOM FEIGELSTOCK AND JUTTA HAUSEN ABSTRACT. We study M-free abelian groups with M- basis X, i.e., each map f : X → M extends uniquely to a homomorphism ϕ : A → M. We will find conditions under which X generates a direct summand of A. If F is an object in a concrete category, X a nonempty set and i : X → F a map, then F is free on the set X if for each M in the category and for each map f : X → M there is a morphism ϕ : F → M such that ϕ◦i = f ,cf.[7]. We will investigate, in the category of abelian groups only, which objects are “free” if, in the above definition, “each M ” is replaced by “some fixed M .” The answer, of course, depends on what kind of abelian group M actually is. Definition. Let A,M be abelian groups and X a subset of A. Then A is M -free with M -basis X if, for each map f : X → M , there is a unique ϕ ∈ Hom(A,M ) such that ϕ ↾ X = f where ϕ ↾ X is the restriction of ϕ to X. We say that A is split-M -free if A = H ⊕〈X〉 such that 〈X〉 is free abelian with basis X and Hom(H,M )=0. Of course, split-M -free implies M -free, and the main purpose of this paper is to investigate for which abelian groups M we have that all M -free groups A are actually split-M -free. Let Cent(R) denote the center of a ring R. We will prove: Main theorem. Let A be M -free with M -basis X and M slender. If either (a) M is countable and End(M ) + , the additive group of the endo- morphism ring of M , is free abelian, or 1991 AMS Mathematics Subject Classification. Primary 20K30. Research of the first author partially supported by Baylor University’s Summer Sabbatical program. Received by the editors on August 13, 2001, and in revised form on October 29, 2001. Copyright c 2002 Rocky Mountain Mathematics Consortium 1367