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A neutrosophic ring is created as a generalization of the fuzzy ring in order to add more properties to the ring theory. In this work, we will study multiple neutrosophic fuzzy rings as a new algebraic generalization of neutrosophic rings. Several types of rings can be studied, such as regular rings, primary rings, and reduced rings. Also, we present the n-refined neutrosophic regular fuzzy subrings, and we give and prove some new results. Finally, we study the concepts related to n-refined neutrosophic reduced fuzzy subrings, and we will obtain new results. [ABSTRACT FROM AUTHOR]

The paper's goal is to examine the Max-ring concept in the present. To achieve our aim, we'll employ a novel notion called a dual module. Greater than the duo module provided by Ozcan and Harmanci is the duo-max-module. We established that a duo-maximal again max-module, Rduo Max-ring, has a direct summand. Also, ℛ is the maximum ring if every module ℳ over ℛ is semi-primitive. We looked at 풩 as the maximum of the max-injective module and ℛ as the duo-max ring. On the other hand, any nonzero element of ℳ has an annihilator that is semimax. Ideally, ℳ is a semi- duo-maximal module, and ℛ is max-ring module. Finally, a number of definitions, attributes, and fresh findings have been presented together with some links between these types of Max-rings. [ABSTRACT FROM AUTHOR]

The main goal of this article is to present new concept of module theory in order to study small submodules based on the Radical of module. Depending on the relevance between small and essential submodules it is observed that every submodule of an R-module M is maximal and essential if and only if the cyclic module xR is not essential small submodule for all x in M. Moreover, a small submodule of a simple module M is essential in M. As well it is showed that every small is essential submodule. We also give more conditions which make a Radical of multiplication and finitely generated R-module largest, essential and small submodule in M. [ABSTRACT FROM AUTHOR]

In this paper, we study and present a new result about retractable module. Several notions are used to obtain more implications like category module, semi-primitive module, non-singular module and their interrelations. Also, we proved that if the radical of M equal to zero, so M is retractable module, and we proved if all maximal submodules have intersection in M and equal zero, this implies that M is retractable modules. Some examples and properties of retractable module has been presented. Finally, every nonsingular module is retractable module. [ABSTRACT FROM AUTHOR]

In this paper, we introduce some new results about the relationship between two concepts namely singular and essential and c-ring. We prove that if any module W is a direct sum of simple submodules imply that the ring T is C-ring. Also, if W has one maximal submodule, this means T is C-ring. We investigate that any module W over T is singular and semisimple imply T is C-ring. Finely, we introduced some new examples and definitions about the topic. [ABSTRACT FROM AUTHOR]

This article provides some new results which shows the basic relationship between some submodules and pseudo valuation modules. As another type of modules we study pseudo – valuation module (P.V.M) by using some submodules such as prime submodule and strongly prime submodule and maximal submodule. We prove that finitely generated module E over the ring W haves prime submodule and this imply that E is P.V.M. Also every module E over fully prime ring is pseudo valuation module. Finally, we introduced L-maximal submodule as a way to study pseudo – valuation module. [ABSTRACT FROM AUTHOR]

If R is a commutative ring containing an identity element. Multiplicative subset Sin R (Mult-(S −1R)), means if 1 ∈ S and ab ∈ S for all a, b ∈ S. In this article, we characterize some particular cases of multiplicative subset in general and we studied the multiplicative subset by using two types of module as projective and flat modules. The main results about multiplicative subsets in module theory have relations with projective and flat module. We proved that if S is a Mult-(S−1R), so S−1R is a flat R-module. Also we obtained if S is a Mult-(S −1R) and P denotes a projective-module, so S−1P is a projective S−1R-module. The other result if S is a Mult-(S −1R), and M a module, then M=S−1M iff M is an S−1R-module. Eventually, we obtained the following result; if S is a multiplicative-subset of a ring R, then S−1R=0 iff S contains a nilpotent element. [ABSTRACT FROM AUTHOR]

In this article, we study several relationships about injective module over Dedekind domain. We prove that every divisible module M over D.V.R is injective. Also, any pseudo injective module with R is a Dedekind leads to M is injective. Several results have been studied over Dedekind domain. [ABSTRACT FROM AUTHOR]

In this article, we introduce some conditions about some concepts in order to obtain injective module ℳ and so any A ≤essℳ(ℳ is, C1−module). Some cases have been studied to approach our objective such as free and anti-hopfian properties. Every submodule of injective module is essential in ℳ and hence ℳ is C1-module. Also, a module ℳ is C1-module if it does not contain proper essential extension, and we will proof every free R-module ℳ1 has 0000 ≤essℳ1. Also, if 0000 = Iℳ and a module ℳ is faithful, then any (0000 ≤essℳ) and so ℳ is C1-module. [ABSTRACT FROM AUTHOR]

The notion of length module was introduced by Lomp (1999) because of utmost importance to the topic of finite length we should be exposed of some modules which it is related to this property. In this article we introduce new conditions over semisimple, simple modules and we discuss some of the basic characterizations of these modules which show many relations between these module and length property. Since any weakly supplemented module with zero radical is semisimple then hdim(M )=length(M ) holds. Therefore supplemented and hollow modules are weakly supplemented with Rad (M )=0 implies hdim(M)=length(M ), therefore since semisimple module is direct sum of simple submodule then any simple module have finite length property. [ABSTRACT FROM AUTHOR]