Let M be a module over a commutative ring R. A submodule P of M is called prime if P≠ M
and, whenever r∈ R, e∈ M, and re∈ P, we have rM⊆ P or e∈ P. We let Spec (M) denote …

We establish conditions for Spec (M) to be Noetherian and spectral space, wrt different
topologies. We used rings with Noetherian spectrum to produce plentiful examples of …

We show that if the power series ring $ R [[X]] $ in one indeterminate over a commutative
ring R with identity is Laskerian, then R is Noetherian. On the other hand, if $ R [[X]] $ is a ZD …

Let R be a commutative ring with identity and let M be an R-module. A proper submodule P
of M is called a classical prime submodule if abm Î P for a, b Î R, and m Î M, implies that am Π…

Introduction: A proper submodule Nofa module M over a ring R is said to be prime or p-
prime if re e N, for re Rand ee M implies that either ee N or rep=(N: M). In [2], Chin-Pi Lu …

Let R be a commutative ring with identity and let M be an R-module. A proper submodule P
of M is called a classical prime submodule if abm∈ P for a, b∈ R, and m∈ M, implies that …

Let $ R $ be a commutative ring and let $ M $ be an $ R $-module. In this article, we
introduce the concept of the Zariski socles of submodules of $ M $ and investigate their …

In this paper we continue our study of classical Zariski topology of modules, that was
introduced in Part I (see [2]). For a left R-module M, the prime spectrum Spec (RM) of M is …

Let M be a non-zero module over an associative (not necessarily commutative) ring. In this
paper, we investigate the so-called second and coprime submodules of M. Moreover, we …

Let R be a commutative ring and M be an R-module. The second spectrum Spec s (M) of M
is the collection of all second submodules of M. We topologize Spec s (M) with Zariski …