Two-rank $\mathrm{SP}$-groups with clean endomorphism rings

The notion of clean ring was introduced by W. K. Nicholson in 1977 to give an example of ring with idempotents, which can be lifted modulo any left (right) ideal. In the case when $R$ is an endomorphism ring of some module, the new descriptions of cleanness property appear which may be useful for study of the conditions of cleanness of ring $R$ [6].
This work continues the author’s investigations of cleanness of endomorphism rings of finite rank $\mathrm{SP}$-groups, which are a class of mixed Abelian groups [2]–[4]. The author earlier prooved cleanness of endomorphism rings of one-rank $\mathrm{SP}$-groups with cyclic $p$-components. Moreover, it was shown that any endomorphism of a finite rank $\mathrm{SP}$-group with cyclic $p$-components is clean if its image is contained in torsion subgroup of $A$.
In this work, the subject of investigation is a two-rank $\mathrm{SP}$-groups with cyclic $p$-components without any infinite torsion direct summand. The proof of cleanness of endomorphism rings of these groups is the key result of this work. Subsidiary statements obtained in proof of this result allows to describe the structure of endomorphism rings of considering groups.