Decomposability and computability

We present a new construction of indecomposable type $\mathbf{0}$ Abelian groups of rank $2$. The new construction is used to study degree spectra of such groups. As a corollary, we obtain a new computability-theoretic proof showing that there exist continuum many nonisomorphic type $\mathbf{0}$ indecomposable Abelian groups of rank $2$.