Let Cc (X) ff PC (X): f (X) is countableg. Similar to C (X) it is observed that the sum of any
collection of semiprime (resp. prime) ideals in the ring Cc (X) is either Cc (X) or a semiprime …

Abstract Let C_c (X) C c (X)(resp. C^ F (X) CF (X)) denote the subring of C (X) C (X)
consisting of functions with countable (resp. finite) image and C_F (X) CF (X) be the socle of …

‎ The author narrates how the study and research on the rings of continuous real-valued
functions on a topological space initiated in Iran‎.‎ In particular‎,‎ almost completely all the …

Let CF (X) be the socle of C (X). It is shown that each prime ideal in C (X)/CF (X) is essential.
For each h∈ C (X), we prove that every prime ideal (resp. zideal) of C (X)/(h) is essential if …

We denote by 𝒜 (R) the class of all Artinian R-modules and by 𝒩 (R) the class of all
Noetherian R-modules. It is shown that 𝒜 (R)⊆ 𝒩 (R)(𝒩 (R)⊆ 𝒜 (R)) if and only if 𝒜 (R/P)⊆ …

In this article we observe that C(X) is integral over I+\mathbbR if and only if I is of the form of
a finite intersection of real maximal ideals. The integral closure of a subring I+\mathbbR is …

Abstract In 2016, Facchini and Nazemian defined the notion of iso-Artinian and iso-
Noetherian modules and rings. We discuss some new properties of iso-Artinian and iso …

The submodules with the property of the title (N⊆ M is strongly essential in M if∏ IN is
essential in∏ IM for any index set I) are introduced and fully investigated. It is shown that for …

The well-known method, which reduces the study of Artinian modules over commutative
rings to the study of Artinian modules over quasi-local rings, is extended to the study of these …

We introduce and study the concept of essential Krull dimension as dual of small Krull
dimension of a module, which are defined in the same vein as Krull dimension of a module …