On extensions of pseudo-integers (Q1325611)

A rank-1 torsion-free abelian group having no \(\infty\) in its type is called a group of pseudo-integers. If \(P\) and \(P'\) are two such rank-1 groups, the author shows (Theorem 4.2) that a rank-2 torsion-free completely decomposable abelian group can occur as a non-splitting extension of \(P\) by \(P'\) exactly when \(\text{type }P < \text{type }P'\). \{Reviewer's remarks: The author is apparently not aware of the classical work of R. Baer on types and completely decomposable groups as reported in \textit{L. Fuchs} [Infinite abelian groups, Vol. II. (1973; Zbl 0257.20035)]. Many of the author's results become easy exercises when formulated in terms of types of rank-1 groups. For example, Theorem 4.1, when reformulated, is a special case of Exercise 1.6(a) in \textit{D. M. Arnold}, Finite rank torsion-free abelian groups and rings (Lect. Notes Math. 931, 1982; Zbl 0493.20034)]. The main theorem (Theorem 4.2) becomes transparent when one looks at the typeset of a rank-2 completely decomposable group\}.