Torsion-Free Modules with $\mathrm{UA}$-Rings of Endomorphisms

An associative ring $R$ is called a unique addition ring ($\mathrm{UA}$-ring) if its multiplicative semigroup $(R,\,\cdot\,)$ can be equipped with a unique binary operation $+$ transforming the triple $(R,\,\cdot\,,+)$ to a ring. An $R$-module $A$ is said to be an $\mathrm{End}$-$\mathrm{UA}$-module if the endomorphism ring $\mathrm{End}_R(A)$ of $A$ is a $\mathrm{UA}$-ring. In the paper, the torsion-free $\mathrm{End}$-$\mathrm{UA}$-modules over commutative Dedekind domains are studied. In some classes of Abelian torsion-free groups, the Abelian groups having $\mathrm{UA}$-endomorphism rings are found.