Separable abelian groups as modules over their endomorphism rings

K. M. Rangaswamy
1984 Proceedings of the American Mathematical Society  
Properly separable mixed abelian groups A which are projective, respectively flat, as modules over their endomorphism rings are completely characterized. These results generalize the works of F. Richman and E. A. Walker. A subclass of separable mixed abelian groups-the properly separable groups-in which finitely many rank one summands can be embedded in a finite rank summand is considered here. It is shown that a reduced properly separable mixed abelian group A is projective as a module over
more » ... endomorphism ring E(A) exactly when A = T @ (0 Bi), i G I, where T is torsion with each of its p-components bounded, for each i G I, Bi contains a torsionfree summand of rank one whose type is idempotent and is the smallest in the typeset of Bl/{Bl)t and, finally, the subgroups T and Bi are all fully invariant in A. A result of F. Richman and E. A. Walker is generalised to show that the reduced properly separable mixed abelian groups A which are E(A)-ñat have the following characterising property: In the typeset of A/At, any nonempty finite subset has a lower bound whenever it has an upper bound.
doi:10.1090/s0002-9939-1984-0740169-3 fatcat:atk245i5w5fjpgofgw6a3an33a