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Let R be an associative and 2-torsion-free ring with an identity. in this work, we will generaliz the results of differentially prime rings in [18] by applying the hypotheses in a differentially semiprime rings. In particular, we have proved that if R is a D-semiprime ring, then either R is a commutative ring or D is a semiprime ring. [ABSTRACT FROM AUTHOR]

A square is a concatenation of two identical words, and a word w is said to have a square yy if w can be written as xyy z for some words x and z. It is known that the ratio of the number of distinct squares in a word to its length is less than two, and any location of a word could begin with two distinct squares which are appearing in the word for the last time. A square whose first location starts with the last occurrence of two distinct squares is an FS-double square. We explore and identify the conditions under which a sequence of locations in a word starts with FS-double squares. We first find the structure of a word that begins with two consecutive FS-double squares and obtain its properties that enable us to extend the sequence of FS-double squares. It is proved that the length of the longest sequence of consecutive FS-double squares in a word of length n is at most nn. We show that the squares in the longest sequence of consecutive FS-double squares are conjugates. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of weakly semiprime ideals and weakly n -systems in noncommutative rings. We establish the equivalence between an ideal P being a weakly semiprime ideal and R − P being a weakly n -system. We provide alternative definitions and explore the properties of weakly semiprime ideals. Additionally, we investigate scenarios where all ideals in a given ring are weakly semiprime and demonstrate that in Noetherian rings, where every ideal is weakly semiprime, the prime radical and the Jacobson radical coincide. [ABSTRACT FROM AUTHOR]

The main objective of this article is to classify all finite singleton local rings, which are associative rings characterized by a unique maximal ideal and a distinguished basis consisting of a single element. These rings are associated with four positive integer invariants p , n , s , and t , where p is a prime number. In particular, we aim to classify these rings and count them up to isomorphism while maintaining the same set of invariants. We have found interesting cases of finite singleton local rings with orders of p 6 and p 7 that hold substantial importance in the field of coding theory. [ABSTRACT FROM AUTHOR]

Let R be an associative ring and M a unitary left R-module. An R-module M is said to be uniserial if its submodules are linearly ordered by inclusion. A serial module is a direct sum of uniserial modules. In this paper, we bring our modest contribution to the open problem listed in the book of Alberto Facchini "Module Theory" which states that: is any direct summand of a serial module serial? The answer is yes for particular rings and R-modules. [ABSTRACT FROM AUTHOR]

Several results about one-sided duo rings and duo rings are generalized from the case of unital rings to the case of arbitrary associative rings in this paper. For example, the characterization of one-sided duo rings and duo rings is given, and it is shown that all idempotents of a duo ring commute with all elements of the ring and that every prime ideal is completely prime. [ABSTRACT FROM AUTHOR]

In this article we introduce and study the concepts of uncountably generated Krull dimension and uncountably generated Noetherian dimension of an R-module, where R is an arbitrary associative ring. These dimensions are ordinal numbers and extend the notion of Krull dimension and Noetherian dimension. They respectively rely on the behavior of descending and ascending chains of uncountably generated submodules. It is proved that a quotient finite dimensional module M has uncountably generated Krull dimension if and only if it has Krull dimension, but the values of these dimensions might differ. Similarly, a quotient finite dimensional module M has uncountably generated Noetherian dimension if and only if it has Noetherian dimension. We also show that the Noetherian dimension of a quotient finite dimensional module M with uncountably generated Noetherian dimension β is less than or equal to ω¹ + β, where ω1 is the first uncountable ordinal number. [ABSTRACT FROM AUTHOR]

In this note, we show that any epimorphism originating at a von Neumann regular ring (not necessary commutative) is a universal localization. As an application, we prove that the Telescope Conjecture holds for the unbounded derived categories of von Neumann regular rings (not necessary commutative). [ABSTRACT FROM AUTHOR]

Assuming that R is an associative ring with prime ideal P, this paper investigates the commutativity of the quotient ring R=P, as well as the possible forms of generalized derivations satisfying certain algebraic identities on R: We give results on strong commutativity, preserving generalized derivations of prime rings, using our theorems. Finally, an example is given to show that the restrictions on the ideal P are not super uous. [ABSTRACT FROM AUTHOR]

We introduce the notion of lifting Baer modules, as a generalization of both Baer and lifting modules and give some of their properties. A module M is called lifting Baer if right annihilator of a left ideal of End(M) lies above a direct summand of M. Also, we define the concepts of r-supplemented and amply r-supplemented modules. It is shown that an amply r-supplemened module M that every supplement submodule, is a direct summand of M, is lifting Baer. The relationships between Baer modules and lifting Baer modules are investigated. Morever, we prove that the endomorphism ring of any lifting Baer module is lifting Baer ring. [ABSTRACT FROM AUTHOR]

The set is a sub-f-ring of RL, that is, the ring of all continuous real-valued functions on a completely regular frame L. The main purpose of this paper is to continue our investigation begun in [3] of extending ring-theoretic properties in RL to the context of completely regular frames by replacing the ring RL with the ring Cc(L) to the context of zero-dimensional frames. We show that a frame L is a CP-frame if and only if Cc(L) is a regular ring if and only if every ideal of Cc(L) is pure if and only if Cc(L) is an Artin-Rees ring if and only if every ideal of Cc(L) with the Artin-Rees property is an Artin-Rees ideal if and only if the factor ring Cc(L)=hi is an Artin-Rees ring for any 2 Cc(L) if and only if every minimal prime ideal of Cc(L) is an Artin-Rees ideal. [ABSTRACT FROM AUTHOR]

Let R be an associative ring with identity and M an R-bimodule. We introduce the generalization of r-clean rings called left r-clean R-bimodules, defined without their endomor-phism rings. An R-bimodule M is said to be left r-clean if each element is the sum of a left idempotent and a left regular element of M. We present some properties of the left r-clean R-bimodule. At the end of this paper, we give the suficient and necessary condition for an R-bimodule to form a left r-clean R-bimodule. [ABSTRACT FROM AUTHOR]

Let R be an associative ring with unity, X be a finite group, H be a subgroup of X, and G be a set of left coset representatives for the left action of H on X. In this article, we introduce two different ways to put R into a non-trivial G -weak graded ring that is a ring graded by the set G which is defined with a binary operation ∗ and satisfying an algebraic structure with specific properties. The first one is by choosing a subset S of G such that S is a group under the ∗ operation and putting R t = 0 for all t ∈ G and t ∉ S . The second way, which is the most important, is induced by combining the operation ∗ defined on G and the coaction ◁ of H on G. Many examples are provided. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the notions uniformly S-flat, briefly u-S-flat, modules and dimensions. We introduce and study the notions of weak u-S-flat modules. An R-module M is said to be weak u-S-flat if Tor1R(R/I,M) is u-S-torsion for any ideal I of R. This new class of modules will be used to characterize u-S-von Neumann regular rings. Hence, we introduce the weak u-S-flat dimensions of modules and rings. The relations between the introduced dimensions and other (classical) homological dimensions are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we will extend some results on the commutativity of Jordan ideals proved in [8] and [5]. However, instead of left generalized derivations, we will consider generalized left derivations, which are sufficient to obtain good results with respect to the structure of near-rings. We will also show that the conditions imposed in the paper cannot be removed. [ABSTRACT FROM AUTHOR]

Let R be an associative ring and let n be a non-negative integer. In this paper, we consider n-super finitely presented, Gorenstein n-weak injective and Gorenstein n-weak flat modules under change of rings. For an excellent extension S ≥ R, we show that the Gorenstein n-super finitely presented dimensions (resp., weak Gorenstein n-super finitely presented dimensions) of rings R and S coincide. [ABSTRACT FROM AUTHOR]

Let R be a ring with an endomorphism α. A ring R is a skew power series McCoy ring if whenever any non-zero power series... We generalized the results of [2] and investigate the relations between the skew power series ring and the standard ring-theoretic properties. Moreover, we obtain some characterizations for skew power series ring R[[x; α]], to be McCoy, zip, strongly AB and has Property (A). [ABSTRACT FROM AUTHOR]

Groups of infinite unitriangular matrices over associative unitary rings are considered. These groups naturally act on infinite dimensional free modules over underlying rings. They are profinite in case underlying rings are finite. Inspired by their connection with groups defined by finite automata the problem to construct faithful representations of free products of groups by banded infinite unitriangular matrices is considered. For arbitrary prime p a sufficient conditions on a finite set of banded infinite unitriangular matrices over unitary associative rings of characteristic p under which they generate the free product of cyclic p-groups is given. The conditions are based on certain properties of the actions on finite dimensional free modules over underlying rings. It is shown that these conditions are satisfied. For arbitrary free product of finite number of cyclic p-groups constructive examples of the sets of infinite unitriangular matrices over unitary associative rings of characteristic p that generate given free product are presented. These infinite matrices are constructed from finite dimensional ones that are nilpotent Jordan blocks. A few open questions concerning properties of presented examples and other types of faithful representations are formulated. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notions of soft quasi-ideal, soft minimal quasi-ideal, soft left (resp. right) N-subgroup and soft invariant subnear-ring of a soft near-ring with the help of soft set theory established by Molodtsov. We also introduce the concepts of soft zero-symmetric near-ring, soft constant near- ring, soft near-field and soft Q- simple near-ring over a near-ring. We investigate the properties of these notions with illustrative examples. We obtain the characterizations of soft quasi-ideals of a soft (zero-symmetric) near-ring and soft nearfields over a near-ring. [ABSTRACT FROM AUTHOR]

With this paper we present an extension of our recent ISSAC paper about computations of Gröbner(-Shirshov) bases over free associative algebras Z 〈 X 〉. We present all the needed proofs in details, add a part on the direct treatment of the ring Z / m Z as well as new examples and applications to e.g. Iwahori-Hecke algebras. The extension of Gröbner bases concept from polynomial algebras over fields to polynomial rings over rings allows to tackle numerous applications, both of theoretical and of practical importance. Gröbner and Gröbner-Shirshov bases can be defined for various non-commutative and even non-associative algebraic structures. We study the case of associative rings and aim at free algebras over principal ideal rings. We concentrate on the case of commutative coefficient rings without zero divisors (i.e. a domain). Even working over Z allows one to do computations, which can be treated as universal for fields of arbitrary characteristic. By using the systematic approach, we revisit the theory and present the algorithms in the implementable form. We show drastic differences in the behaviour of Gröbner bases between free algebras and algebras, which are close to commutative. Even the process of the formation of critical pairs has to be reengineered, together with implementing the criteria for their quick discarding. We present an implementation of algorithms in the Singular subsystem called Letterplace , which internally uses Letterplace techniques (and Letterplace Gröbner bases), due to La Scala and Levandovskyy. Interesting examples and applications accompany our presentation. [ABSTRACT FROM AUTHOR]

Let R be a G graded commutative ring and M be a G-graded R-module. The set of all graded second submodules of M is denoted by S pecsG(M); and it is called the graded second spectrum of M. We discuss graded rings with Noetherian graded prime spectrum. In addition, we introduce the notion of the graded Zariski socle of graded submodules and explore their properties. We also investigate S pecsG(M) with the Zariski topology from the viewpoint of being a Noetherian space. [ABSTRACT FROM AUTHOR]

In this paper, we modify the result of M. N. Daif [1] on multiplicative derivations in rings. He showed that the multiplicative derivation is additive by imposing certain conditions on the ring <. Here, we have proved the above result with lesser conditions than M. N. Daif for getting multiplicative derivation to be additive. [ABSTRACT FROM AUTHOR]

The note concerns the commutativity of associative rings (possibly nonunital) endowed with a derivation. Our focus is on δ-prime rings. We give a new proof of Hirano and Tominaga's result that a δ-prime ring is commutative whenever the derivation δ is nonzero and commuting on a nonzero two-sided δ-ideal. We also provide some further generalizations of Herstein's classical theorem on a prime ring admitting a nonzero derivation with commutative range. [ABSTRACT FROM AUTHOR]

Let R !A be a homomorphism of associative rings, and let .F ;C/be a hereditary complete cotorsion pair in R-Mod. Let .FA;CA/be the cotorsion pair in A-Mod in which FA is the class of all left A-modules whose underlying R-modules belong to F. Assuming that the F -resolution dimension of every left R-module is finite and the class F is preserved by the coinduction functor HomR.A;􀀀/, we show that CA is the class of all direct summands of left A-modules finitely (co)filtered by A-modules coinduced from R-modules from C. Assuming that the class F is closed under countable products and preserved by the functor HomR.A;􀀀/,we prove that CA is the class of all direct summands of leftA-modules cofiltered by A-modules coinduced from R-modules from C, with the decreasing filtration indexed by the natural numbers. A combined result, based on the assumption that countable products of modules from F have finite F -resolution dimension bounded by k, involves cofiltrations indexed by the ordinal ! Ck. The dual results also hold, provable by the same technique going back to the author's monograph on semi-infinite homological algebra (2010). In addition, we discuss the n-cotilting and n-tilting cotorsion pairs, for which we obtain better results using a suitable version of a classical Bongartz-Ringel lemma. As an illustration of the main results of the paper, we consider certain cotorsion pairs related to the contraderived and coderived categories of curved DG-modules. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity, M an R-module and E a torsion-free R-module. A submodule N of M is said to be essential (large) in M if the intersection of N with each nonzero submodule of M is nonzero, that is, N... Rm ≠ 0 for any nonzero element m € M and we write N ≤e M. It is clear that the class of e - exact sequences is larger than the class of exact sequences. In this study we present the concept of e-injective modules as a generalization of injective modules. The main goal is to give a characterization of e-injective modules in terms of contravariant functor Hom(−,E). [ABSTRACT FROM AUTHOR]

A local ring is an associative ring with unique maximal ideal. We associate with each Artinian local ring with singleton basis four invariants (positive integers) p , n , s , t. The purpose of this article is to describe the structure of such rings and classify them (up to isomorphism) with the same invariants. Every local ring with singleton basis can be constructed over its coefficient subring by a certain polynomial called the associated polynomial. These polynomials play significant role in the enumeration. [ABSTRACT FROM AUTHOR]

Unlike for polynomial rings, the notion of multiplication for the near-ring of polynomials is the substitution operation. This leads to somewhat surprising results. Let S be an abelian left near-ring with identity. The relation ~ on S defined by letting a ~ b if and only if annS(a) = annS(b), is an equivalence relation. The compressed zero-divisor graph ΓE(S) of S is the undirected graph whose vertices are the equivalence classes induced by ~ on S other than [0]S and [1]S, in which two distinct vertices [a]S and [b]S are adjacent if and only if ab = 0 or ba = 0. In this paper, we are interested in studying the compressed zero-divisor graphs of the zero-symmetric near-ring of polynomials R0[x] and the near-ring of the power series R0[[x]] over a commutative ring R. Also, we give a complete characterization of the diameter of these two graphs. It is natural to try to find the relationship between diam(ΓE(R0[x])) and diam(ΓE(R0[[x]])). As a corollary, it is shown that for a reduced ring R, diam(ΓE(R)) ≤ diam(ΓE(R0[x])) ≤ diam(ΓE(R0[[x]])). [ABSTRACT FROM AUTHOR]

A module M is called a V -module (cosemisimple module) if every simple module in σ[M] is M-injective. A module M is an SSI-module if every semisimple module in σ[M] is M-injective. The main objective of this paper is to prove that a module M is an SSI-module if and only if M is a locally noetherian V -module. We also prove that an R-module M over a commutative ring R is semisimple if and only if every hereditary pretorsion class in ptors-M is an idempotent radical. [ABSTRACT FROM AUTHOR]

In this paper, we characterise weakly Gorenstein injective and weakly Gorenstein coflat comodules over triangular matrix coalgebras by introducing the class of weakly compatible bicomodules. In particular, Gorenstein injective and Gorenstein coflat comodules are investigated. [ABSTRACT FROM AUTHOR]

Suppose that N is a near-ring and P is a 3-prime ideal of N. In this paper we introduce the notion of (α, τ)-P derivation in near-rings, we also study the structure of the quotient near-ring N=P which satisfies certain algebraic identities involving (α, τ)-P derivations. [ABSTRACT FROM AUTHOR]

An associative ring with unity is called exchange if every element is exchange, i.e., there exists an idempotent e ∈ aR such that 1-e ∈ (1-a)R; if this representation is unique for every element, we call the ring uniquely exchange. We give a complete description of uniquely exchange rings. [ABSTRACT FROM AUTHOR]

Let R be a prime ring of characteristic different from 2 with a right Martindale quotient ring Qr and an extended centroid C. Let F be a non zero generalized derivation of R and S be the set of evaluations of a non central valued multilinear polynomial f(x1, ..., xn) over C. Let p, q ∈ R be such that pF²(u)u + F²(u)uq = 0 for all u ∈ S. Then for all x ∈holds: 1. there exists a ∈ Qr is central valued on R and there exists a ∈ Qr such that F(x) = ax with pa² + a²q = 0. [ABSTRACT FROM AUTHOR]

One of the four basic machine learning tasks is pattern classification. The selection of the proper learning algorithm for a given problem is a challenging task, formally known as the algorithm selection problem (ASP). In particular, we are interested in the behavior of the associative classifiers derived from Alpha-Beta models applied to the financial field. In this paper, the behavior of four associative classifiers was studied: the One-Hot version of the Hybrid Associative Classifier with Translation (CHAT-OHM), the Extended Gamma (EG), the Naïve Associative Classifier (NAC), and the Assisted Classification for Imbalanced Datasets (ACID). To establish the performance, we used the area under the curve (AUC), F-score, and geometric mean measures. The four classifiers were applied over 11 datasets from the financial area. Then, the performance of each one was analyzed, considering their correlation with the measures of data complexity, corresponding to six categories based on specific aspects of the datasets: feature, linearity, neighborhood, network, dimensionality, and class imbalance. The correlations that arise between the measures of complexity of the datasets and the measures of performance of the associative classifiers are established; these results are expressed with Spearman's Rho coefficient. The experimental results correctly indicated correlations between data complexity measures and the performance of the associative classifiers. [ABSTRACT FROM AUTHOR]

Let R be an associative ring with identity. In this paper we extend the J.H. Maynes results, which he treatised in [27]. In particular, we prove that if R is a δ-prime ring with char R ≠ 2 and I is a nonzero δ-ideal of R, where 0 ≠ δ ∈ D, c ∈ R and [c, δ(c)] in the center of R, then R is commutative. [ABSTRACT FROM AUTHOR]

Let R be a commutative von Neumann regular ring. We show that every finitely generated ideal I in the ring of polynomials R[X] has a strong Gröbner basis. We prove this result using only the defining property of a von Neumann regular ring. [ABSTRACT FROM AUTHOR]

The theory of neutrosophic multi fuzzy ideals of γ near ring is dispensed in this work and various algebraic properties such as intersection, union of neutrosophic multi fuzzy ideals of γ near ring are examined. [ABSTRACT FROM AUTHOR]

Childhood stunting remains a major public health issue in many low- and middle-income countries. In Ghana, the progress made is insufficient to reach the targets set by the 2025 World Health Assembly and the 2030 United Nations Sustainable Development Goals. Although studies have examined the socio-demographic determinants of childhood stunting, there has not been any systematic study to examine the spatial associative effects of the socio-demographic and socio-ecological factors at the district level, where health programmes are implemented and monitored. Bayesian geo-additive semiparametric regression technique was used to analyse five conservative rounds of Demographic and Health Surveys in Ghana, with socio-ecological covariates derived from the Demographic and Health Survey Program Geospatial Covariate datasets to examine the temporal trends in childhood stunting, the extent of geospatial clustering at the district level and their associative relationships with socio-demographic and socio-ecological factors. The findings show that childhood stunting in Ghana is not spatially randomly distributed but clustered. Clustering of high childhood stunting was observed amongst districts in the Upper West, Upper East, Northern, North East, Savannah, and Western North regions, whilst clustering of low childhood stunting was observed in districts in the Greater Accra, Volta, Bono and the Eastern regions. Whist socio-demographic factors were predominantly associated with clustering of districts with high childhood stunting, the socio-ecological factors were mainly associated with clustering of districts with low childhood stunting. The socio-ecological factors identified to have a nonlinear associative effect with childhood stunting were Insecticide Treated Net (ITN) coverage, nightlight composite, travel time to a main settlement and population density. The findings suggest that targeted interventions at the district level are essential to reducing childhood stunting in Ghana. [ABSTRACT FROM AUTHOR]

The theory of anti neutrosophic multi fuzzy ideals of γ near ring is dispensed in this work and various algebraic properties such as intersection, union of anti neutrosophic multi fuzzy ideals of γ near ring are examined. Further we examined the direct anti product of anti neutrsophic multi fuzzy ideals of near ring and also we proved the homorphic images and pre images of anti neutrosophic multi fuzy ideals of γ near ring. [ABSTRACT FROM AUTHOR]

A module M is called s -coseparable if for every nonzero submodule U of M such that M/U is finitely generated, there exists a nonzero direct summand V of M such that V ⊆ U and M/V is finitely generated. It is shown that every non-finitely generated free module is s -coseparable but a finitely generated free module is not, in general, s -coseparable. We prove that the class of s -coseparable modules over a right noetherian ring is closed under finite direct sums. We show that the class of commutative rings R for which every cyclic R-module is s -coseparable is exactly that of von Neumann regular rings. Some examples of modules M for which every direct summand of M is s -coseparable are provided. [ABSTRACT FROM AUTHOR]

Let M be an R-module. If every essential submodule of M has a g-radical supplement in M, then M is called an essential g-radical supplemented (or briefly eg-radical supplemented) module. In this work, some properties of these modules are investigated. It is proved that every factor module and every homomorphic image of an eg-radical supplemented module are eg-radical supplemented. Let M be an eg-radical supplemented module. Then every finitely M-generated R-module is eg-radical supplemented. [ABSTRACT FROM AUTHOR]

A moduleM is called ss-lifting if for every submodule A of M, there is a decomposition M = M1 ⊕ M2 such that M1 ≤ A and A∩M2 ⊆ Socs (M), where Socs(M) = Soc(M)∩Rad(M). In this paper, we provide the basic properties of ss-lifting modules. It is shown that: (1) a module M is ss-lifting iff it is amply ss-supplemented and its ss-supplement submodules are direct summand; (2) for a ring R, RR is ss-lifting if and only if it is ss-supplemented iff it is semiperfect and its radical is semisimple; (3) a ring R is a left and right artinian serial ring and Rad (R) ⊆ Soc(RR) iff every left R-module is ss-lifting. We also study on factor modules of ss-lifting modules. [ABSTRACT FROM AUTHOR]

Let 1 < k and m, k 2 Z+. In this manuscript, we analyse the action of (semi)-prime rings satisfying certain differential identities on some suitable subset of rings. To be more specific, we discuss the behaviour of the semiprime ring R satisfying the differential identities ([d([s, t]m), [s, t]m])k = [d([s, t]m), [s, t]m] for every s, t 2 R. [ABSTRACT FROM AUTHOR]

The element q of a ring R is called quasi-idempotent element if q2 = uq for some central unit u of R, or equivalently q = ue, where u is a central unit and e is an idempotent of R. In this paper, we define that the ring R is almost quasi-clean if each element of R is the sum of a regular element and a quasi-idempotent element. Several properties of almost-quasi clean rings are investigated. We prove that every quasi-continuous and nonsingular ring is almost quasi-clean. Finally, it is determined that the conditions under which the idealization of an R-module M is almost quasi clean. [ABSTRACT FROM AUTHOR]

Most of the research on weakly prime and weakly 2-absorbing modules is for modules over commutative rings. Only scatterd results about these notions with regard to non-commutative rings are available. The motivation of this paper is to show that many results for the commutative case also hold in the non-commutative case. Let R be a non-commutative ring with identity. We define the notions of a weakly prime and a weakly 2-absorbing submodules of R and show that in the case that R commutative, the definition of a weakly 2-absorbing submodule coincides with the original definition of A. Darani and F. Soheilnia. We give an example to show that in general these two notions are different. The notion of a weakly m-system is introduced and the weakly prime radical is characterized interms of weakly m-systems. Many properties of weakly prime submodules and weakly 2-absorbing submodules are proved which are similar to the results for commutative rings. Amongst these results we show that for a proper submodule Ni of an Ri-module Mi, for i = 1; 2, if N1 × N2 is a weakly 2-absorbing submodule of M1 × M2, then Ni is a weakly 2-absorbing submodule of Mi for i = 1, 2. [ABSTRACT FROM AUTHOR]

Let R be a ring with identity 1R. A subset J of R is called a γ-set if for every a ∈ R\J, there exist b, c ∈ J such that a+b = 0 and ac = 1R = ca. A γ-set of minimum cardinality is called a minimum γ-set. In this study, we identified some elements of R that are necessarily in a γ-sets, and we presented a method of constructing a new γ-set. Moreover, we gave: necessary and sufficient conditions for rings to have a unique γ-set; an upper bound for the total number of minimum γ-sets in a division ring; a lower bound for the total number of minimum γ-sets in a division ring; necessary and sufficient conditions for T(x) and T to be equal; necessary and sufficient conditions for a ring to have a trivial γ-set; necessary and sufficient conditions for an image of a γ-set to be a γ-set also; necessary and sufficient conditions for a ring to have a trivial γ-set; and, necessary and sufficient conditions for the families of γ-sets of two division rings to be isomorphic. [ABSTRACT FROM AUTHOR]

We introduce the notion of a modied near module M over a near ring N and explain a method of obtaining near ring multiplications via a special type of maps from M into N called semilinear maps. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity and N(R) and J (R) denote the nilradical and the Jacobson radical of R, respectively. A proper ideal I of R is called an n-ideal if for every a, b ∈ R, whenever ab ∈ I and a ∉ N(R), then b ∈ I. In this paper, we introduce and study J-ideals as a new generalization of n-ideals in commutative rings. A proper ideal I of R is called a J-ideal if whenever ab ∈ I with a ∉ J (R), then b ∈ I for every a, b ∈ R. We study many properties and examples of such class of ideals. Moreover, we investigate its relation with some other classes of ideals such as r-ideals, prime, primary and maximal ideals. Finally, we, more generally, define and study J-submodules of an R-modules M. We clarify some of their properties especially in the case of multiplication modules. [ABSTRACT FROM AUTHOR]

A ringed partially ordered set with zero is a pair (L,F), where L is a partially ordered set with a least element 0L and F:L → Ring is a covariant functor. Here the partially ordered set L is given a category structure in the usual way and Ring denotes the category of associative rings with identity. Let RingedParOrd0 be the category of ringed partially ordered sets with zero. There is a functor H:Ring → RingedParOrd0 that associates to any ring R a ringed partially ordered set with zero (Hom(R),FR). The functor H has a left inverse Z:RingedParOrd0 → Ring. The category RingedParOrd0 is a fibred category. [ABSTRACT FROM AUTHOR]