Lifting modules and their various generalizations as some main concepts in module theory have been studied and investigated extensively in recent decades. Some authors tried to present some homological aspects of lifting modules and -supplemented modules. In this work, we shall present a homological approach to -supplemented modules via fully invariant submodules. Lifting modules and H-suppleme...

A group Γ is defined to be cofinitely Hopfian if every homomorphism Γ → Γ whose image is of finite index is an automorphism. Geometrically significant groups enjoying this property include certain relatively hyperbolic groups and many lattices. A knot group is cofinitely Hopfian if and only if the knot is not a torus knot. A free-by-cyclic group is cofinitely Hopfian if and only if it has trivi...

Lifting modules plays important roles in module theory. H-supplemented are a nice generalization of lifting which have been studied extensively recently. In this article, we introduce proper via images fully invariant submodules. Let F be submodule right Rmodule M. We say that M is IF -H-supplemented case for every endomorphism φ M, there direct summand D such φ(F) + X = if and only It proved d...

Let F be a global field of characteristic p > 0, F/F a Galois extension with Gal(F/F ) ≃ Z l (for some prime l 6= p) and E/F a non-isotrivial elliptic curve. We study the behaviour of Selmer groups SelE(L)r (r any prime) as L varies through the subextensions of F via appropriate versions of Mazur’s Control Theorem. With mild hypotheses on SelE(F )r (essentially a consequence of the Birch and Sw...

In this paper we provide various properties of Rad-⊕-supplemented modules. In particular, we prove that a projective module M is Rad⊕-supplemented if and only if M is ⊕-supplemented, and then we show that a commutative ring R is an artinian serial ring if and only if every left R-module is Rad-⊕-supplemented. Moreover, every left R-module has the property (P ∗) if and only if R is an artinian s...

Let R be ring and M a right R-module. This article introduces the concept of τ −⊕-supplemented modules as follows: Given a hereditary torsion theory in Mod-R with associated torsion functor τ we say that a module M is τ −⊕-supplemented when for every submodule N of M there exists a direct summand K of M such that M = N +K and N ∩K is τ−torsion, and M is called completely τ −⊕-supplemented if ev...