Elementary equivalence of endomorphism rings and automorphism groups of periodic Abelian groups

In this paper, we prove that the endomorphism rings EndA and EndA^′ of periodic infinite Abelian groups A and A^′ are elementarily equivalent if and only if the endomorphism rings of their p-components are elementarily equivalent for all primes p. Additionally, we show that the automorphism groups AutA and AutA^′ of periodic Abelian groups A and A^′ that do not have 2-components and do not contain cocyclic p-components are elementarily equivalent if and only if, for any prime p, the corresponding p-components A_p and A_p^′ of A and A^′ are equivalent in second-order logic if they are not reduced, and are equivalent in second-order logic bounded by the cardinalities of their basic subgroups if they are reduced. According to [11], for such groups A and A^′, their automorphism groups are elementarily equivalent if and only if their endomorphism rings are elementarily equivalent, and the automorphism groups of the corresponding p-components for all primes p are elementarily equivalent.