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 …

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 …

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…

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 …

Inspired by the interplay between the Zariski topology defined on the prime spectrum of a
commutative ring R and the ring theoretic properties of R in [3, 8, 18, 23, 25], we introduce in …

It is shown that the well-known characterizations of when a commutative ring R has
Noetherian spectrum carry over to characterizations of when the set Spec (M) of prime …

Let R be a commutative ring and M an R-module. The purpose of this article is to introduce a
new class of modules over R called X-injective R-modules, where X is the prime spectrum of …

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 …

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. We examine the
situation where for each prime ideal ρof R the set of all ρ-prime submodules of M is finite. In …