Closed ss -semilocal modules and rings

A module A is designated as closed ss -semilocal module provided for any closed submodule G of A , there exists a submodule H of A such that A = G + H and G ∩ H ≤ Soc s ( A ) where Soc s ( A ) = Rad ( A ) ∩ Soc ( A ) and a ring S is named as closed ss -semilocal ring provided the left S -module S S is a closed ss -semilocal module. Any direct summand of a closed ss -semilocal module is also closed ss -semilocal. A finite sum of closed ss -semilocal modules is a closed ss -semilocal, in case this sum is a local distributive module. Over a closed ss -semilocal ring S , any nonsingular cyclic S -module is a closed ss -semilocal module. A free S -module A is a closed ss -semilocal module if and only if the endomorphism ring of A , End S ( A ) is a closed ss -semilocal module. A ring S is a closed ss -semilocal ring if and only if for any positive integer n , the ring of all n × n matrices over S , M n ( S ) is a closed ss -semilocal ring. For a refinable module A , if there is a closed submodule E of A such that E = G + K for some K ≤ Soc s ( A ) and for any G ≤ A , then A is a ⊕ ss -supplemented module if and only if A is an ss -supplemented module if and only if A is an ss -semilocal module if and only if A is a closed ss -semilocal module.