SOME INJECTIVE CLASSES

All groups are assumed to be in the category of torsion free abelian groups of finite rank with homomorphisms. Let TF be this category of groups. We consider three natural methods of generating injective classes of groups. One method produces all cotorison theories in TF which we characterize. The work done is similar to L. Salce's work. Other work done is similar to work done by W. Wickless and C. Vinsonhaler.

We use these injective classes to prove the following. Let I(C(H)) denote the injective class generated by H. We prove: (a) The reduced groups in I(C({G})) are G-projective if G (TURNEQ) X(,1) (CRPLUS)...(CRPLUS) X(,n) where each X(,i) is rank 1 and if i (NOT=) j then {p (VBAR) pX(,i) (NOT=) X(,i)} (INTERSECT) {p (VBAR) pX(,j) (NOT=) X(,j)} = (SLASHCIRC). (b)

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is the class of groups Y, Y (TURNEQ) X(,n(,1)) (CRPLUS)...(CRPLUS) X(,n(,k)) (CRPLUS) Q('n) for some integers n(,1),...,n(,k),n, if

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is a set of rank 1 groups with {p (VBAR) pX(,j) (NOT=) X(,j)} (INTERSECT) {p (VBAR) pX(,i) (NOT=) X(,i)} = (SLASHCIRC) if i (NOT=) j.