Extending modules are generalizations of injective modules and, dually, lifting modules
generalize projective supplemented modules. There is a certain asymmetry in this duality …

Let R be a ring, M be a left R-module and Spec (RM) be the collection of all prime
submodules of M. In this paper and its sequel, we introduce and study a generalization of …

For a fixed natural number m≥ 2, an element a of a ring R is m-nil clean if a= e+ b where
em= e∈ R and b is nilpotent; if further eb= be, a is called strongly m-nil clean. The ring R is …

A ring R is (strongly) weakly nil clean if every element in R is the sum or difference of a
nilpotent and an idempotent (that commutes). In this note, we show that if R is strongly nil …

In this article, we introduce and study a generalization of the classical Krull dimension for a
module MR. This is defined to be the length of the longest strong chain of prime submodules …

We carry out a study of rings R for which Hom_R(M,N)≠0 for all nonzero N≦M_R. Such
rings are called retractable. For a retractable ring, Artinian condition and having Krull …

A right module M over a ring R is said to be retractable if Hom R (M, N)≠ 0 for each nonzero
submodule N of M. We show that M⊗ R RG is a retractable RG-module if and only if MR is …

This paper describes properties of three certain classes of rings determined by conditions on
idempotents and units, namely, the condition that any two generators of each principal right …

A ring is right tall if every non-noetherian right module contains a proper non-noetherian
submodule. We prove a ring-theoretical criterion of tall commutative rings. Besides other …

A sufficient condition is given for a ring to be either strongly regular or left self-injective
regular with non-zero socle. If A is a left self-injective ring such that the left annihilator of …