A category equivalence between homogeneous completely decomposable abelian groups and modules over principal ideal domains

AbstractLet Z⊆R be a subgroup of the rationals and (S,≤) a finite poset. In this paper we introduce the category Rep(S,R) of homogeneous completely decomposable groups H of type R with distinguished homogeneous completely decomposable subgroups Hi(i∈S) of the same type, respecting the order of S, i.e. if i,j∈S and i≤j, then Hi⊆Hj. We construct a category equivalence between the two categories Rep(S,R) and Rep(S,R0), where R0=End(R). Using this equivalence we are able to obtain decomposition theorems for certain subclasses of Rep2(R) and Rep3(R). We prove that these special representations admit a decomposition into indecomposable representations of rank ≤2