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Let R be a associative ring with identity. The problem of characterizing rings, based on the property that every element in R is the sum of two elements with specific attributes, is a major area of study in ring theory. Hirano and Tominaga (1988) showed that if R is a ring in which every element is the sum of two commuting idempotents, then every element in R is tripotent. Following that, Ying et al. (2016) gave some characterizations of the ring R where every element is the sum of two commuting tripotent elements. In general, characterizing R such that every element in R is a sum of two commuting m-potent for a given positive integer m, remains an open problem. This paper will provide some answers to the aforementioned problem over the 픽qCn group ring, where 픽q is a finite field of order q and Cn is a finite cyclic group or order n. In conclusion, this paper provides a general method to compute the characteristic of 픽qCn group ring such that every element is a sum of two commuting m-potent. Furthermore, the characteristic of the 픽qCn group ring can be used to characterize the group ring as a group ring that has Sm,2 property, thereby providing additional properties about 픽qCn. [ABSTRACT FROM AUTHOR]

In recent times, spherical fuzzy sets have come to the forefront of researchers interest and have been used for algebraic structures such as groups, rings and near rings. In this paper we propose the spherical fuzzy ideals in near rings. Moreover we prove the union of spherical fuzzy ideals is also spherical fuzzy ideal. Also we prove the intersection of spherical fuzzy ideals is also spherical fuzzy ideal. Also we combine the rough set and spherical fuzzy sets. Then we will prove union of two rough spherical fuzzy sets is also a rough spherical fuzzy set. Next, we will present union of two rough spherical fuzzy sets is also a rough spherical fuzzy set. To illustrate these concepts some examples are established [ABSTRACT FROM AUTHOR]

We presented the novel concept of cubic strong bi-ideals of cubic near-rings in this paper, which is a broader approach to fuzzy strong bi-ideals of near-rings. We also looked into a couple of its characteristics using examples. [ABSTRACT FROM AUTHOR]

Let 픽 be a field of any characteristic. For any associative algebra A over 픽, we can define a Lie algebra by defining a Lie product [a, b]=ab−ba for any two elements a and b in A, where ab is the associative product in A. Moreover, if A is equipped with an involution ∗: A→A, then the space of the skew symmetric elements K={a∈A ∣ a∗=−a} is a Lie subalgebra of A. It is natural to except that the Lie algebra that obtained in this way has a structure which is closely connected with the associative structure of A. In this paper, we study the relation between simple associative algebras and their related finitary symplectic and finitary orthogonal Lie algebras. [ABSTRACT FROM AUTHOR]

As a dual of essential submodule , Takechi defines a coessential submodule. Let Ε be a unitary left R-module on associative ring R with identity , a submodule 퐅 of Ε is called coessential submodule of a submodule 퐋 in Ε where 퐅≤퐋≤횬 ,if 퐋 퐅 ≪ 횬 퐅 .Also Golar defines a coclosed submodule provided it has noproper coessential submodule in Ε. A submodule 퐅 of an R-module E is called supplement of 퐕 퐢퐟 퐄=퐅+퐕 퐚퐧퐝 퐅∩퐕≪퐅 , an R-module is called supplemented if every submodule of E is supplement. In this paper we study Ζ-Coessential submodule and Ζ-Coclosed submodule as generalizations of coessential and coclosed submodules. We give many properties related with these types of submodules. [ABSTRACT FROM AUTHOR]

An element x in a ring R with unity is called almost clean if there exists some idempotent element e in R and a regular element u in R such that x = e + u. This article aims to show the existence of almost clean elements in a certain matrix ring over an integral domain R with unity and provide its characterizations. [ABSTRACT FROM AUTHOR]

In this piece of work, we establish and study the concept of a L-VPI-GNR and it's properties, taking values in a complete Brouwerian lattice satisfying infinite meet distributive law. Also, we determine all L-VPI-GNR of a GNR, Q by establishing a one-to-one correspondence between L-VPI-GNR of Q and the pairs (I, δ) where I is a prime ideal of Q and 'δ ' is a prime element in L. [ABSTRACT FROM AUTHOR]

The idea of Boolean ring has been generalized in a variety of ways by several Mathematicians. Von-Neumann regular rings, p-rings, pk−rings, Boolean–like rings , Associative rings and P1 & P2−rings are a few interesting ring theoretic generalizations of Boolean rings. Boolean-like rings developed by A.L. Foster [3] preserve and confirm to the formal properties of the Boolean ring. V. Swaminathan [4] studied the structure of Boolean-like rings and established new results regarding Boolean-like rings besides extending many of the results true for Boolean rings to Boolean-like rings. The concept of Boolean- like algebra is introduced. It is proved that a Boolean-like algebra is a Boolean algebra if and only if aΛa = a for all a. Some results pertaining to the mutual implications of Boolean-like algebra and Boolean-like ring are presented in this paper. As in the case of Boolean algebra and Boolean ring, it is proved that the Boolean-like algebra and Boolean-like ring are equivalent abstract structures. [ABSTRACT FROM AUTHOR]

In this piece of research work, we Originate and analyse the notion of a Lattice-vague maximal ideal of a Γ-Near ring and explore various properties of it taking values in a complete Brouwerian lattice satisfying infinite meet distributive law. [ABSTRACT FROM AUTHOR]

In this research article, we propose and explore the idea of a L-VI-GNR and several properties of it, taking values in a complete lattice L- satisfying infinite meet distributive law. Also, we demonstrate that the intersection of two L-VI-GNR is also a L-VI-GNR. Furthermore, we discussed the particular case of union of two L-VI-GNR is a L-VI-GNR. [ABSTRACT FROM AUTHOR]

In this paper we wish to introduce the concept of tripolar fuzzy weak bi-ideal of a near ring and study some of its properties. [ABSTRACT FROM AUTHOR]

Bipolar fuzzy set is the generalization of fuzzy set. Based on the bipolar fuzzy set theory, the concept of bipolar fuzzy quasi ideals and bipolar N subgroups of near rings are introduced. Pertaining to this notion we also introduced some basic definitions, theoretical properties and characterizations of bipolar fuzzy quasi ideals. [ABSTRACT FROM AUTHOR]

A ring R with unity is called clean if every element x ∈ R can be written as a sum of an idempotent and a unit element in ring R. Meanwhile, a ring with unity is called feebly clean, if every element r in the ring can be written as r = u + e − f, with ef = fe = 0, where u is a unit element in the ring and e, f is an idempotent element in the ring. [7] shows the existence of clean elements in a subring X3(D) of the matrix ring 3 × 3 over an integral domain D. Based on these results, we provide the set of all clean elements in the matrix ring X3(ℤ), we show the existence of feebly clean elements in the matrix ring X3(ℤ), and show the connections between those elements and feebly clean elements in the matrix ring X3(ℤ). These connections are different from [10]. [ABSTRACT FROM AUTHOR]

In this research article, we introduce the notion of sum and direct sum of vague ideals of a near-ring and developed some results related to the vague ideals of a near-ring with suitable example. [ABSTRACT FROM AUTHOR]

Author introduce the notion of a (δ, γ)-fuzzy bi-ideal of a near-ring and obtain the characterization of a bi-ideal in terms of a (δ, γ)-fuzzy bi-ideal of a near-ring. We establish that every (δ, γ)-fuzzy left (resp. right) N-subgroup or (δ, γ)-fuzzy left (resp. right) ideal of a near-ring is a (δ, γ)-fuzzy bi-ideal of a near-ring. But the converse is not neceassrily true. Further, we discuss the properties of (δ, γ)-fuzzy bi-ideal of a near-ring. [ABSTRACT FROM AUTHOR]

In this article I continue the study of Sub-Stable Near-Rings, introduced in my earlier papers. If a subset F ≠ {0} of N satisfies xF = xFx = Fx for all x ∈ N then F is known as a stabilizing subset of N. An attempt is made in this article to study the properties of Sub-Stable Near-Rings when it is regular. It is observed that set of all F-Subgroups in a regular Sub-Stable Near-Ring is a Boolean Algebra under certain conditions. [ABSTRACT FROM AUTHOR]

In this paper, we defined minC11 and maxC11 modules M over associative ring R with identity and find out further properties, characterization of direct sum of minC11 and maxC11 modules and some related basic properties. The main results were any direct sum of modules with minC11 satisfies minC11 and also for maxC11. In particular we give the conditions for the direct summand of minC11-module is minC11-module. Moreover for a commutative ring R, if M is a faithful, finitely generating and multiplication R-module, then M satisfies minC11 (maxC11) if and only if R satisfies minC11 (maxC11). [ABSTRACT FROM AUTHOR]

In the present paper, we prove that 3-prime near-ring N is commutative ring, if any one of the following conditions are satisfied: (i) f (N) ⊆ Z, (ii) f ([x,y]) = 0, (iii) f ([x,y]) = ±τ([x,y]), (iv) f ([x,y]) = ±τ (xoy), (v) f([x,y]) = τ ([d(x),y]), for all x, y ∈ N, where f is a nonzero left multiplicative generalized (σ, τ)-derivation of N associated with a multiplicative (σ, τ)- derivation d. [ABSTRACT FROM AUTHOR]

In this paper, we construct some topologies on a module M by using the dual Zariski topology on a quotient module of M and define a continuous map between these topologies. Then we find a topology on M which is homeomorphic to the dual Zariski topology on a quotient module of M. [ABSTRACT FROM AUTHOR]

Let μ be a special class of rings, i.e. class of prime associative rings which is closed under ideals and essential extensions. Let *μ be the class of all *μ-rings, that is, rings in μ such that the factor ring is not in μ for every nonzero ideal of R. Apart from their interesting properties, *μ-rings from a generalization of the important class of *-rings introduced and studied in [4], [7], [8], and [14]. France-Jackson and Roux proved useful in describing minimal elements of lattices of hereditary radicals [9]. It was observed in [9] that some special classes μ of rings, such as the class of all simple rings with unity, for example, coincide with *μ and the question was asked to describe all such special classes. We will give a partial answer to this question by noting that, in fact, this holds for every special class of simple rings with unity. However, the converse is not true. [ABSTRACT FROM AUTHOR]

Let R be an associative ring with identity and let n be a positive integer. An element x∈ R is n -clean if x = u1 +...+ un + e for some units ui ∈ R (i = 1,..., n) and idempotent e∈R . The ring R is said to be n -clean if all of its elements are n -clean. We say that an element x∈R is n -weakly clean if x = u1 +...+un + e or x = u1 +...+un -e for some unit ui ∈ R (i =1,...,n) and idempotent e ∈ R . The ring R is said to be n -weakly clean if all of its elements are n -weakly clean. In this paper we obtain some properties of n -weakly clean rings. [ABSTRACT FROM AUTHOR]

Let R be an associative ring with identity. An element x ∈ R is clean if x can be written as the sum of an idempotent element and a unit in R. The ring R is said to be clean if all of its elements are clean. An element x ∈ R is weakly clean if x can be written as x = u+e or x = u-e for some unit u and idempotent e in R. The ring R is said to be weakly clean if all of its elements are weakly clean. Let n be a positive integer. An element x ∈ R is n -clean if x can be written as the sum of an idempotent and n units in R. The ring R is said to be n -clean if all of its elements are n -clean. In this paper we extend some results on clean rings to weakly clean rings and n -clean rings. [ABSTRACT FROM AUTHOR]