Modules over Endomorphism Rings

It is proved that $A$ is a right distributive ring if and only if all quasiinjective right $A$-modules are Bezout left modules over their endomorphism rings if and only if for any quasiinjective right $A$-module $M$ which is a Bezout left $\operatorname{End}(M)$-module, every direct summand $N$ of $M$ is a Bezout $\operatorname{End}(M)$-module. If $A$ is a right or left perfect ring, then all right $A$-modules are Bezout left modules over their endomorphism rings if and only if all right $A$-modules are distributive left modules over their endomorphism rings if and only if $A$ is a distributive ring.