‎ 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 …

Our aim is to introduce a ring of some discontinuous real-valued functions defined on a
topological space X. By C (X) F we denote the set of all real-valued functions on the …

Several characterizations of a ring R is given for which any non-zero finitely generated
module M is RETRACTABLE in the sense that HomR (M, N) is non-zero whenever N is a …

An ideal L of a ring R is said to be strongly hollow (briefly, sh-ideal) if L⊆ A+ B implies that
either L⊆ A or L⊆ B, for every pair A and B of ideals of R. We first, study and investigate …

We observe that every non-commutative unital ring has at least three maximal commutative
subrings. In particular, non-commutative rings (resp., finite non-commutative rings) in which …

Consider the ring $ C_c (X) _F $ of real valued functions which are discontinuous on a finite
set with countable range. We discuss $(\mathcal {Z} _c) _F $-filters on $ X $ and $(\mathcal …

We call a submodule K of an R-module M strongly superfluous in M provided that for any
index set I, is a superfluous submodule of. The basic properties of strongly superfluous …

Let 𝑅 be a ring with identity. Recall that a submodule 𝑁 of a left 𝑅-module 𝑀 is called
strongly essential if for any nonzero subset 𝑋 of 𝑀, there is 𝑟∈ 𝑅 such that (0≠) 𝑟𝑋⊆ 𝑁 …

In the present paper, modules over integral domains and principal ideal domains that are
proper essential extensions of some submodules are classified. We introduce a new class of …

The submodules with the property of the title (a submodule $ N $ of an $ R $-module $ M $
is called strongly dense in $ M $, denoted by $ N\leq_ {sd} M $, if for any index set $ I …