This paper investigates when the group algebra \(kG\) of a countable soluble group with trivial periodic radical is primitive. In such groups there is a characteristic torsion-free abelian subgroup \(B\), the Zalesskii subgroup, with the property that an ideal \(I\) of \(kG\) is zero if and only if \(I\cap kB\) is. A result of K. A. Brown [Arch. Math. 36, 404–413 (1981; Zbl 0464.16009)] allows the discussion to rest on constructing faithful simple modules in the case where \(B\) is locally of finite \(\mathbb Z\)-rank as a \(\mathbb Z\langle x\rangle\)-module for all \(x\) in \(G\). At this point the work of the author [Math. Proc. Camb. Philos. Soc. 97, 27–49 (1985; Zbl 0532.16007)] is appealed to. An example of the type of result obtained is:
Theorem B. Let \(G\) be a finitely generated soluble group with trivial periodic radical and let \(k\) be a field. (i) If \(k\) is uncountable then \(kG\) is primitive if and only if \(\Delta(G)=1\). (ii) If \(k\) is countable then \(kG\) is primitive if either (a) there exists a torsion-free abelian subgroup \(A\) of infinite rank such that \(A\cap \Delta(G)=[A,B]=1\), or (b) \(k\) is non-absolute and \(\Delta(G)=1\).