Commutative group rings of finite representation type
Let \(p\) be a rational prime and let \(\mathbb{Z}_ p\) be the localisation of \(\mathbb{Z}\) at \(p\). Let \(G\) be a finite abelian group of finite representation type, i.e. \(G\) has at most finitely many isomorphism classes of indecomposable \(\mathbb{Z}_ p G\)-lattices; the \(p\)-Sylow subgroup of \(G\) is necessarily cyclic of order at most \(p^ 2\). This paper determines the possible ranks of such \(\mathbb{Z}_ p G\)-lattices. It is shown that an indecomposable \(\mathbb{Z}_ p G\)-lattice may be embedded as a sublattice of \(\mathbb{Z}_ p G^{(4)}\) but that, if the commutativity of \(G\) is abandoned, the corresponding conclusion, even in a weaker sense, is false, thereby contradicting the author's own previous remark [J. Algebra 129, 26-74 (1990; Zbl 0688.20003)].
indecomposable \(\mathbb{Z}_ p G\)-lattices
scientific article; zbMATH DE number 89580
scientific article