\({\mathbb{Z}}G_ n\)-modules, \(G_ n\) cyclic of square-free order n
As the title suggests, the aim of this paper is to study finitely generated modules over the integral group ring \(R={\mathbb{Z}}G\) of a cyclic group G of square-free order n. Such a ring R is Dedekind-like in the following sense: there exist positive integers s, t, Dedekind domains \(R_ i\) (1\(\leq i\leq s)\) which are not fields, fields \(K_ j\) (1\(\leq j\leq t)\) and ring homomorphisms \(f,g : \tilde R=\oplus^{s}_{i=1}R_ i\to \oplus^{t}_{j=1}K_ j\) such that \(\ker f+\ker g=\tilde R\) and R is the pullback \(\{r\in \tilde R : f(r)=g(r)\}.\) Modules over Dedekind-like rings have been extensively studied by the author in two earlier papers [J. Algebra 71, 62-114 (1981; Zbl 0508.16009) and 93, 1- 116 (1985; Zbl 0564.13010)] so that the present paper can be viewed as a continuation of these earlier investigations. Let \(R={\mathbb{Z}}G\) as above. On the whole finitely generated R-modules are well behaved. For example, the torsion-free rank of every indecomposable finitely generated R-module is at most n and every torsion element is annihilated by a power of n. Let P be a maximal ideal of R and \(R_ P\) the P-localized group ring (see the author's 1985 paper). Then in common with all Dedekind-like rings R has the property that if M is an indecomposable finitely generated R-module then \(M_ P\) is the direct sum of at most two indecomposable \(R_ P\)-modules. On the other hand, unlike Dedekind-like rings in general, finitely generated \(R_ P\)-modules satisfy the Krull- Schmidt theorem. In startling contrast, for every integer \(m\geq 2\) there exist a positive integer n and an R-module M such that M is free Abelian of finite rank (as a \({\mathbb{Z}}\)-module) but M is the direct sum of k indecomposable R-modules for all \(2\leq k\leq m\). Finally, if \(n\geq 2\) is a square-free integer and X a finite non-zero Abelian group which is annihilated by a power of n and which has no direct summands of prime order then the Abelian group \(R\oplus X\) can be given the structure of a faithful indecomposable R-module of projective dimension 1 and this can be done in at least \(| Pic R|\) non-isomorphic ways.
finitely generated modules over the integral group ring
scientific article; zbMATH DE number 3959598