Totally Co-hopfian Modules

In the present paper, a right R-module M is said to be totally co-Hopfian if any f ∈ EndR(M) wite this property that, for each positive integer n, ker f n is a superfluous submodule of M, is an epimorphism. The class of totally co-Hopfian modules lies properly between the class of strongly co-Hopfian modules and the class of co-Hopfian modules. We show that over endoregular modules, the concepts of the Hopficity, co-Hopficity, totally Hopficity and totally co-Hopficity are coincide. Moreover, over commuta_x0002_tive domains, torsion free totally co-Hopfian and divisible totally Hopfian modules (Abelian groups) are characterized. A Theorem of Utumi, [6, Corollary 1.5], that characterized the essential right ideals of a right self-injective ring, is generalized to right quasi-injective modules and by this we show that a right quasi-injective R-module M is totally Hopfian if and only if S = EndR(M) is a left totally co-Hopfian ring. Consequently, a right self-injective ring R is righ totally Hopfian if and only if R is a left totally co-Hopfian ring.