$SP$-groups with clean endomorphism rings

The notion of a clean ring was introduced by W. K. Nicholson in 1977 as an example of a ring with idempotents, that can be lifted modulo any left (right) ideal [1]. The class of clean rings is a proper subclass of the class of exchange rings [1, 2].
In the case when $R$ is an endomorphism ring of some module, new descriptions of the cleanness property appear. They can be useful for the study of conditions for cleanness of a ring $R$. This subject recently attracted attention of many specialists [5, 6].
In this work, some aspects of cleanness of endomorphism rings of $SP$-groups are considered. These groups are one of classes of mixed Abelian groups [7, 8]. The cleanness of endomorphism rings of self-small $SP$-groups is proved. Some sufficient conditions are found for the converse proposition to hold. The structure of endomorphism rings of rank one $SP$-groups with cyclic $p$-groups is described and their cleanness is proved, the description of Jacobson radical of endomorphism rings of such groups is found. Some sufficient conditions of cleanness for endomorphisms of finite rank $SP$-groups with cyclic $p$-groups are obtained.