An algebra is called endoprimal if only its term functions permute with all of its endomorphisms. In [1] we solved, in particular, the endoprimality problem for the class of abelian groups that split into a direct sum of a torsionfree group of rank 1 and a torsion group. Here we generalise that result to arbitrary abelian groups of torsionfree rank 1, that is, to such groups whose quotient by the torsion part T is of rank 1. Given a mixed abelian group A, its nucleus N is a subring of the field of rational numbers Q generated by the inverses of prime numbers p such that the p-component Tp of A is zero and the quotient A/T is p-divisible. Clearly the group A has a natural structure of an N-module. When studying endoprimality of abelian groups, it is conveniemt to consider them as modules over their nucleus and actually to study the endoprimality of those modules. Our main result is the following.
Theorem. Let A be an abelian group of torsionfree rank 1 with nucleus N, torsion part T and p-components Tp. Denote by P the set of primes p such that A/T is p-divisible. The group A is endoprimal if and only if T is unbounded and for every p in P one of the following three conditions is satisfied: 1) Tp=0, 2) Tp is not reduced, 3) Tp is not a direct summand of A.
[1] Kaarli, K. and Márki, L., Endoprimal abelian groups. J. Austral. Math. Soc. (Series A) 67 (1999), 412-428.