LAUSR

We consider a representation of quasi-endomorphisms of Abelian torsion-free groups of rank 4 bymatrices of order 4 over the field of rational numbers Q. We obtain a classification for quasi-endomorphism rings of Abelian torsion-free groups of rank 4 quasi-decomposable into a direct sum of groups A1, A2 of rank 1 and strongly indecomposable group B of rank 2 such that quasi-homomorphism groups Q ⊗ Hom(Ai, B) and Q ⊗ Hom(B, Ai) for any i = 1, 2 have rank 1 or are zero. Moreover, for algebras from the classification we present necessary and sufficient conditions for their realization as quasi-endomorphism rings of these groups.