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

Let image 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 (iset membership, variantS) of the same type, respecting the order of S, i.e. if i,jset membership, variantS and i≤j, then Hisubset of or equal toHj. 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.