Totally Hopfian and totally co-Hopfian modules

In this paper, we define a right [Formula: see text]-module [Formula: see text] to be totally Hopfian (respectively, totally co-Hopfian) if every endomorphism [Formula: see text] with this property satisfies that, for each positive integer [Formula: see text], the image of [Formula: see text] (respectively, the kernel of [Formula: see text]) is an essential (respectively, superfluous) submodule of [Formula: see text], with [Formula: see text] being a monomorphism (respectively, an epimorphism). The class of totally Hopfian (respectively, totally co-Hopfian) modules is properly situated between the classes of strongly Hopfian (respectively, strongly co-Hopfian) modules and Hopfian (respectively, co-Hopfian) modules. In addition to other results, this study demonstrates that, for endoregular modules, the concepts of hopficity, co-hopficity, totally hopficity, and totally co-hopficity are equivalent. Furthermore, over commutative domains, torsion-free totally co-Hopfian and divisible totally Hopfian modules are characterized. The theorem of Utumi, [Y. Utumi, Self-injective rings, J. Algebra 6 (1967) 56–64, Corollary 1.5], which characterizes the essential right ideals of a right self-injective ring, is generalized to right quasi-injective modules. This generalization allows us to show that a right quasi-injective [Formula: see text]-module [Formula: see text] is totally Hopfian if and only if [Formula: see text] is a left totally co-Hopfian ring. Consequently, a right self-injective ring [Formula: see text] is right totally Hopfian if and only if [Formula: see text] is a left totally co-Hopfian ring.