strongly noncosingular modules

an r-module m is called strongly noncosingular if it has no nonzero rad-small (cosingular) homomorphic image in the sense of harada. it is proven that (1) an r-module m is strongly noncosingular if and only if m is coatomic and noncosingular; (2) a right perfect ring r is artinian hereditary serial if and only if the class of injective modules coincides with the class of (strongly) noncosingular r-modules; (3)absolutely coneat modules are strongly noncosingular if and only if r is a right max-ring and injective modules are strongly noncosingular; (4) a commutative ring r is semisimple if and only if the class of injective modules coincides with the class of strongly noncosingular r-modules.