Generating cotorsion theories and injective classes
Let H be a class of torsion-free abelian groups of finite rank (hereafter, simply ``groups''). This paper considers three classes containing H; (1) Let \(P(H)\) (respectively, \(I(H)\)) be the class of all groups Y satisfying \(Ext(Y,X)=0\) (respectively, \(Ext(X,Y)=0)\) for all X in H. (2) Let \(C(H)\) be the class of groups A such that each X in H is injective with respect to every pure embedding \(A\to B\) of groups. Define \(IC(H)\) to be the class of all groups injective with respect to each pure embedding \(A\to B\) of groups with A in C(H). (3) Let \(E(H)\) be the class of all exact sequences of groups relative to which each X in H is injective and let \(IE(H)\) be the class of all groups Y such that Y is injective with respect to each sequence in \(E(H)\). Clearly, \(H\subset I(P(H)) \cap IC(H) \cap IE(H)\). In fact, \(IE(H)\subset IC(H) \cap I(P(H))\). The author proves a number of other results about these classes, including the following characterization of IP(H). Theorem 4. The class \(IP(H)\) is the class of all groups Y satisfying (a) \(\{p\) prime \(|\;pY\neq Y\}\subset \cup_{X\in H}\{p|\;pX\neq X\}\) and (b) If J is an infinite subset of \(\{p|\) \(pY\neq Y\}\), then there is an X in H with \(J\cap \{p|\) \(pX\neq X\}\) infinite.
torsion-free abelian groups of finite rank
scientific article; zbMATH DE number 4047957
scientific article