Search

The Drazin inverse is connected with the notion of index and core-nilpotent decomposition whenever it is discussed in the context of ring of matrices over complex field. In the absence of Drazin inverse for a given element from an arbitrary associative ring (not necessarily with unity), in this paper, the notion of right (left) core-nilpotent decomposition has been introduced and established its relations with right (left) π -regular property. In fact, the class of such decomposition has been characterized. In case of regular ring, observed that an element is right (left) π -regular if and only if it has a right (left) core-nilpotent decomposition. In the process, several properties of sharp order in an associative ring are studied and with the help of the same, new characterizations of Drazin inverse over an associative ring are obtained and the relation between core-nilpotent decomposition and the Drazin inverse is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of ring called w p - S -Noetherian ring, which is a weak version of S -Noetherian ring property and study the transfer of this notion to various context of commutative ring extensions such as direct product, trivial ring extensions and amalgamation of rings. Furthermore, we define the concept of nonnil w p - S -Noetherian ring property which is a generalization of the w p - S -Noetherian domain property and establish a characterization of this notion using pullbacks. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity and S be a multiplicatively closed subset of R. The purpose of this paper is to introduce the concept of weakly S -primary ideals as a new generalization of weakly primary ideals. An ideal I of R disjoint with S is called a weakly S -primary ideal if there exists s ∈ S such that whenever 0 ≠ a b ∈ I for a , b ∈ R , then s a ∈ I or s b ∈ I. The relationships among S -prime, S -primary, weakly S -primary and S - n -ideals are investigated. For an element r in any general ZPI-ring, the (weakly) S r -primary ideals are characterized where S r = { 1 , r , r 2 , ... }. Several properties, characterizations and examples concerning weakly S -primary ideals are presented. The stability of this new concept with respect to various ring-theoretic constructions such as the trivial ring extension and the amalgamation of rings along an ideal are studied. Furthermore, weakly S -decomposable ideals and S -weakly Laskerian rings which are generalizations of S -decomposable ideals and S -Laskerian rings are introduced. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a generalization of coretractable modules. An R -module M is called coretractable relative to an R-module N if Hom R (M / X , N) ≠ 0 , for any proper submodule X of M. It is shown that a right R -module M is coretractable relative to ⊕ i ∈ I N i if and only if M is coretractable relative to N i , for some i ∈ I. Also we show that if R is a right quasi-duo ring, then there exists a right R -module that it is coretractable relative to N , for any nonzero right R -module N if and only if R is a left perfect local ring. [ABSTRACT FROM AUTHOR]

Let G and A be finite groups of relative coprime orders and A act on G via automorphisms. In this paper, we prove that when every maximal A-invariant subgroup of G that contains the normalizer of some Sylow subgroup has prime index, then G is supersolvable; if every non-nilpotent maximal A-invariant subgroup of G has prime index or is normal in G, then G is a Sylow tower group. [ABSTRACT FROM AUTHOR]

Let G be a group with identity e and R be a graded multiplicative hyperring. In this paper, we introduce and study the notion of graded n -absorbing hyperideals of R as a generalization of prime hyperideals. Many properties, examples and characterizations of graded n -absorbing hyperideals of R are introduced. Moreover, we investigate the behavior of this structure under homogeneous components, graded hyperring homomorphisms, Cartesian product and fundamental relations. We consider the fundamental relation γ ∗ on R as the smallest equivalence relation such that the quotient R / γ ∗ is a ring. Also, we show that if R is a graded hyperring, then R / γ ∗ is a graded ring. [ABSTRACT FROM AUTHOR]

Semiabelian rings, defined by the property that each of their idempotents is either left semicentral or right semicentral, are one among several natural generalizations of abelian rings. In this semi-expository paper, we review a number of interesting properties of semiabelian (and other closely allied) rings that are so far well known only for abelian rings. For instance, semiabelian rings R are always "J-abelian" in the sense that each idempotent of R maps onto a central idempotent in R /rad(R). On the other hand, J-abelian rings turn out to be precisely the "strongly perspective rings" as well as the "strongly IC rings", and the von Neumann regular elements in such rings are automatically strongly regular and are closed under taking n -th powers. In addition, all J-abelian exchange rings have idempotent stable range one, and are in particular clean (although not necessarily strongly clean) rings. [ABSTRACT FROM AUTHOR]

We provide three generalizations of all best known generating sets for S n and A n : (1) m -cycles with consecutive entries; (2) m -cycles with fixed first ℓ entries; (3) two cycles, either (1 , ... , m) , (1 , ... , n) or (1 , ... , m) , (2 , ... , n). Our work on (3) fills a gap in the history of the problem, going back to a 1939 paper by S. Piccard [Sur les bases du groupe symétrique et du groupe alternant, Math. Ann. 116(1) (1939) 752–767]. [ABSTRACT FROM AUTHOR]

It is well known that Frobenius groups can be defined by their complement subgroups. But until now we cannot use a complement subgroup to define a modular Frobenius group. In the present paper, a generalization of Frobenius complements is used as a characterization of a class of modular Frobenius groups. In fact, we build a connection between modular Frobenius groups and Frobenius–Wielandt groups. [ABSTRACT FROM AUTHOR]

The aim of the paper is to characterize the equation 1 + Δ (R) = U (R) , where J (R) ⊆ Δ (R) = { x ∈ R : x + u ∈ U (R)   for all   u ∈ U (R) } that is the largest Jacobson radical subring of R and U (R) is the set of invertible elements of a ring R. We show that this equation is closely related to U J -rings and rings whose elements can be written as the sum of an idempotent and an element from Δ (R). After presenting several characterizations and properties of this equation, we consider the rings satisfying the equation 1 + Δ (R) = U (R) within many well-studied classes of rings. Finally, we close the paper with group rings. [ABSTRACT FROM AUTHOR]

Extriangulated categories were introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. In this paper, we show that the idempotent completion of an extriangulated category admits a natural extriangulated structure. As an application, we prove that the idempotent completion of a recollement of extriangulated categories is still a recollement. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring and S a given multiplicative closed subset of R. In this paper, we introduce the new concept of S -torsion exact sequences (respectively, S -torsion commutative diagrams) as a generalization of exact sequences (respectively, commutative diagrams). As an application, they can be used to characterize two classes of modules that are generalizations of projective modules. [ABSTRACT FROM AUTHOR]

Let S be a commutative semiring with unity different than zero. In this paper, we study the concept of 2-absorbing ideals of S which can be considered as a generalization of prime ideals. Among others, it is shown that the radical of a 2-absorbing ideal is also a 2-absorbing ideal and there are at most 2 prime k-ideals of S that are minimal over a 2-absorbing ideals. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of polyominoes, called closed paths, and we study the primality of their associated ideal. Inspired by an existing conjecture that characterizes the primality of a polyomino ideal by nonexistence of zig-zag walks, we classify all closed paths which do not contain zig-zag walks, and we give opportune toric representations of the associated ideals. To support the conjecture, we prove that having no zig-zag walks is a necessary and sufficient condition for the primality of the associated ideal of a closed path. Finally, we present some classes of prime polyominoes viewed as generalizations of closed paths. [ABSTRACT FROM AUTHOR]

Let D be an integral domain and w be the so-called w-operation on D. In this paper, we define the notion of w-ZPUI domains which is a natural generalization of ZPUI domains introduced by Olberding in 2000. We say that D is a w-ZPUI domain if every nonzero proper w-ideal I of D can be written as I = (J P 1 ⋯ P n) w for some w-invertible ideal J of D and { P 1 , ... , P n } is a nonempty collection of pairwise w-comaximal prime w-ideals of D. Then, among other things, we show that D is a w-ZPUI domain if and only if the polynomial ring D [ X ] is a w-ZPUI domain, if and only if D is a strongly discrete independent ring of Krull type. We construct three types of new w-ZPUI domains from an old one by A + X B [ X ] -construction, pullback, and D [ { X i , Y i , U i , V i } ] / ({ X i V i − Y i U i }) -domains. We also show that given an abelian group G, there is a ZPUI domain with ideal class group G but not a Dedekind domain. Finally, we introduce and study the notion of w-ISP domains as a generalization of w-ZPUI domains. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of classical 2-prime subsemimodules of a semimodule M over a commutative semiring S with identity, which is a generalization of classical prime subsemimodules. A proper subsemimodule N of M having the property that for each a , b ∈ S and each subsemimodule K of M, the inclusion a b K ⊆ N implies a 2 K ⊆ N or b 2 K ⊆ N is called classical 2-prime. We give some results concerning classical 2-prime subsemimodules. Also, the classical 2-prime avoidance theorem for subsemimodules is proved. Finally, the notion of valuation semimodules is introduced and some results on classical 2-prime subsemimodules in valuation semimodules are given. [ABSTRACT FROM AUTHOR]

The investigation of orthodox P -restriction semigroups was initiated by Jones in 2014 as generalizations of orthodox * -semigroups. The aim of this paper is to further study orthodox P -restriction semigroups based on the known results of Jones. After establishing a construction theorem for orthodox P -restriction semigroups, we introduce proper P -restriction semigroups (which are necessarily orthodox) and prove that every (finite) orthodox P -restriction semigroup has a (finite) proper cover. Our results enrich and extend existing results for restriction semigroups and orthodox * -semigroups. [ABSTRACT FROM AUTHOR]

Let R be a semiprime ring, not necessarily with unity, with extended centroid C. For a ∈ R , let T (a) (respectively I (a) , Ref (a)) denote the set of all outer (respectively inner, reflexive) inverses of a in R. In the paper, we study the inclusion properties of T (a) , I (a) and Ref (a). Among other results, we prove that for a , b ∈ R with a von Neumann regular, Ref (a) ⊆ T (b) (respectively Ref (a) ⊆ I (b)) if and only if a = E [ a ] b (respectively I (a) ⊆ I (b)). Here, E [ a ] is the smallest idempotent in C such that E [ a ] a = a. This gives a common generalization of several known results. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity, S be a multiplicatively closed subset of R , and M be an R -module. A submodule N of M is called coidempotent if N = ((0 : M Ann R 2 (N)). Also, M is called fully coidempotent if every submodule of M is coidempotent. In this paper, we introduce the concepts of S -coidempotent submodules and fully S -coidempotent R -modules as generalizations of coidempotent submodules and fully coidempotent R -modules. We explore some basic properties of these classes of R -modules. [ABSTRACT FROM AUTHOR]

We introduce the notion of semi-poor modules and consider the possibility that all modules are either injective or semi-poor. This notion gives a generalization of poor modules that have minimal injectivity domain. A module M is called semi-poor if whenever it is N -injective and N ≠ 0 , then the module N has nonzero socle. In this paper the properties of semi-poor modules are investigated and are used to characterize various families of rings. We introduce the rings over which every module is either semi-poor or injective and call such condition property (P). The structure of the rings that have the property (P) is completely determined. Also, we give some characterizations of rings with the property (P) in the language of the lattice of hereditary pretorsion classes over a given ring. It is proved that a ring R has the property (P) iff either R is right semi-Artinian or R = R 1 × R 2 where R 1 is a semisimple Artinian ring and R 2 is right strongly prime and a right c -ring with zero right socle. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a class of commutative rings which is a generalization of ZD -rings and rings with Noetherian spectrum. A ring R is called strongly J -Noetherian whenever the ring R r is J -Noetherian for every non-nilpotent r ∈ R. We give some characterizations for strongly J -Noetherian rings and, among the other results, we show that if R [ [ x ] ] is strongly J -Noetherian, then R has Noetherian spectrum, which is a generalization of Theorem 2 in Gilmer and Heinzer [The Laskerian property, power series rings, and Noetherian spectra, Proc. Amer. Math. Soc.79 (1980) 13–16]. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity and S a multiplicative subset of R. In this paper, the notion of S -pseudo-irreducible ideals of R as a generalization of pseudo-irreducible ideals is introduced and some related properties are investigated. Then, we consider the decomposition of proper ideals of a ring into a product of pairwise comaximal S -pseudo-irreducible ideals. Some other results are obtained too. [ABSTRACT FROM AUTHOR]

A right R -module M is called a U -module if, whenever A and B are submodules of M with A ≅ B and A ∩ B = 0 , there exist two direct summands K and T of M such that A ⊆ ess K , B ⊆ ess T and K ⊕ T ⊆ ⊕ M. The class of U -modules is a strict and simultaneous generalization of quasi-continuous, square-free and automorphism-invariant modules. In this paper we study the rings whose cyclics are U -modules, extending many of the known results on the subject and providing new ones. [ABSTRACT FROM AUTHOR]

Type B semigroups are generalizations of inverse semigroups, and every inverse semigroup admits an E -unitary cover (M. Petrich, Inverse Semigroups (Wiley, New York, 1984)). Motivated by studying E -unitary cover for inverse semigroups, and as a continuation of Petrich's works in inverse semigroups, in this paper, we first introduce the concept of ∗-prehomomorphism of a type B semigroup. After obtaining some basic properties, we get some structure theorems and give some conditions for a type B semigroup which is constructed by using the ∗-prehomomorphism to be proper. In particular, we introduce the notion of E -unitary good cover for an abundant semigroup, and prove that every type B semigroup with compatible natural partial order admits an E -unitary good cover. [ABSTRACT FROM AUTHOR]

A formation 𝔉 of finite groups has the generalized Wielandt property for residuals, or 𝔉 is a GWP-formation, if the 𝔉 -residual of a group generated by two 𝔉 -subnormal subgroups is the subgroup generated by their 𝔉 -residuals. The main result of this paper describes a large family of GWP-formations to further the transparence of this kind of formations, and it can be regarded as a natural step toward the solution of the classification problem. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity. In this paper, we introduce the concept of 1 -absorbing prime ideals which is a generalization of prime ideals. A proper ideal I of R is called 1 -absorbing prime if for all nonunit elements a , b , c ∈ R such that a b c ∈ I , then either a b ∈ I or c ∈ I. Some properties of 1 -absorbing prime are studied. For instance, it is shown that if R admits a 1 -absorbing prime ideal that is not a prime ideal, then R is a quasi–local ring. Among other things, it is proved that a proper ideal I of R is 1 -absorbing prime if and only if the inclusion I 1 I 2 I 3 ⊆ I for some proper ideals I 1 , I 2 , I 3 of R implies that I 1 I 2 ⊆ I or I 3 ⊆ I. Also, 1 -absorbing prime ideals of PIDs, valuation domains, Prufer domains and idealization of a modules are characterized. Finally, an analogous to the Prime Avoidance Theorem and some applications of this theorem are given. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to introduce and study quasi-order Γ -semihypergroups that is a generalization of quasi-order semihypergroups. Also, we introduce quasi-order semihypergroups associated with quasi-order Γ -semihypergroups and some important results obtained. Moreover, we study complete parts and fundamental relation on quasi-order Γ -semihypergroups and prove that the fundamental relation is transitive. [ABSTRACT FROM AUTHOR]

Let R be a commutative semiring with nonzero identity and M an R -semimodule. In this paper, we introduce the concept of ϕ -primal subsemimodule of M that is a generalization of primal ideal of a commutative ring. Then we give some examples and properties of these subsemimodules. Also, some characterizations of ϕ -primal subsemimodules are presented. [ABSTRACT FROM AUTHOR]

This paper focusses on the question of when modules have ADS- preenvelopes and covers. For a ring , it is proved that every ADS module over is injective if and only if every right -module has an ADS-(pre)envelope (an ADS-(pre)cover). In this paper, we also introduce a generalization of ADS modules stated in terms of their invariance under certain automorphisms of their envelopes. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a generalized inverse of a matrix, namely, the generalized WG inverse, which a generalization of the WG inverse. Several necessary and sufficient conditions such that a matrix to be generalized WG invertible are obtained. Moreover, the formulae of generalized WG inverse of a matrix are given. Finally, we give a algebraic form of the generalized WG inverse. [ABSTRACT FROM AUTHOR]

In this paper, we present a dual version of T. Brzeziński's results about Rota–Baxter systems which appeared in [Rota–Baxter systems, dendriform algebras and covariant bialgebras, J. Algebra460 (2016) 1–25]. Then as a generalization to bialgebras, we introduce the notion of Rota–Baxter bisystem and construct various examples of Rota–Baxter bialgebras and bisystems in dimensions 2, 3 and 4. On the other hand, we introduce a new type of bialgebras (named mixed bialgebras) which consist of an associative algebra and a coassociative coalgebra satisfying the compatible condition determined by two coderivations. We investigate coquasitriangular mixed bialgebras and the particular case of coquasitriangular infinitesimal bialgebras, where we give the double construction. Also, we show in some cases that Rota–Baxter cosystems can be obtained from a coquasitriangular mixed bialgebras. [ABSTRACT FROM AUTHOR]

Hereditary rings have been extensively investigated in the literature after Kaplansky introduced them in the earliest 50's. In this paper, we study the notion of a Σ -Rickart module by utilizing the endomorphism ring of a module and using the recent notion of a Rickart module, as a module theoretic analogue of a right hereditary ring. A module M is called Σ -Rickart if every direct sum of copies of M is Rickart. It is shown that any direct summand and any direct sum of copies of a Σ -Rickart module are Σ -Rickart modules. We also provide generalizations in a module theoretic setting of the most common results of hereditary rings: a ring R is right hereditary if and only if every submodule of any projective right R -module is projective if and only if every factor module of any injective right R -module is injective. Also, we have a characterization of a finitely generated Σ -Rickart module in terms of its endomorphism ring. Examples which delineate the concepts and results are provided. [ABSTRACT FROM AUTHOR]

In this paper, we study the atomic structure of the family of Puiseux monoids, i.e. the additive submonoids of . Puiseux monoids are a natural generalization of numerical semigroups, which have been actively studied since mid-19th century. Unlike numerical semigroups, the family of Puiseux monoids contains non-finitely generated representatives. Even more interesting is that there are many Puiseux monoids which are not even atomic. We delve into these situations, describing, in particular, a vast collection of commutative cancellative monoids containing no atoms. On the other hand, we find several characterization criteria which force Puiseux monoids to be atomic. Finally, we classify the atomic subfamily of strongly bounded Puiseux monoids over a finite set of primes. [ABSTRACT FROM AUTHOR]

The zero-divisor graphs of modules introduced and studied in [S. Safaeeyan, M. Baziar and E. Momtahan, A generalization of the zero-divisor graph for modules, J. Korean Math. Soc. 51(1) (2014) 87-98]. Basic results for zero-divisor graphs of -modules were obtained in [M. Baziar, E. Momtahan and S. Safaeeyan, Zero-divisor graph of abelian groups, J. Algebra Appl. 13(6) (2014) 13]. In this paper, zero-divisor graphs in the title are studied. Here, among other things, we generalize results stated in [M. Baziar, E. Momtahan and S. Safaeeyan, Zero-divisor graph of abelian groups, J. Algebra Appl. 13(6) (2014) 13]. Some results for modules over non-integral domains are also obtained. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with 1 ≠ 0 and n a positive integer. The main purpose of this paper is to study the concepts of n -irreducible and strongly n -irreducible ideals which are generalizations of irreducible and strongly irreducible ideals, respectively. A proper ideal I of R is called n -irreducible (respectively, strongly n -irreducible) if for each ideals I 1 , ... , I n + 1 of R , I = I 1 ∩ ⋯ ∩ I n + 1 (respectively, I 1 ∩ ⋯ ∩ I n + 1 ⊆ I) implies that there are n of the I i 's whose intersection is I (respectively, whose intersection is in I). [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of partial groupoid actions on R -semicategories as well as we give criteria for existence of a globalization of it. This point of view is a generalization of the notions of partial groupoid actions on rings and partial group action on an R -semicategory. We also define the notions of partial skew groupoid category, smash product and describe functorial relations between them, in particular we show that the smash product is a Galois covering of its associated skew groupoid category. [ABSTRACT FROM AUTHOR]

Let (, ) be a finite-dimensional Hom-Hopf algebra. In this paper we mainly construct the Drinfel'd double in the setting of Hom-Hopf algebras by two ways, one of which generalizes Majid's bicrossproduct for Hopf algebras (see [S. Majid, Foundations of Quantum Group Theory (Cambridge University Press, 1995)]) and another one is to introduce the notion of dual pairs of Hom-Hopf algebras. Then we study the relation between the Drinfel'd double and Heisenberg double , generalizing the main result in [J. H. Lu, On the Drinfel'd double and the Heisenberg double of a Hopf algebra, Duke Math. J. 74 (1994) 763-776]. The examples given in the paper are especially, not obtained from the usual Hopf algebras. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to investigate two characteristic subgroups ω풜(G) and θ풜(G) of a finite group G, which are defined as the intersections of the normalizers of derived subgroups of subnormal and non-subnormal subgroups of G respectively. Our main theory improve and extend some earlier results. [ABSTRACT FROM AUTHOR]

Let f : A → B , g : A → C be ring homomorphisms and let 𝔟 (respectively, 𝔠) be an ideal of B (respectively, C) satisfying f − 1 (𝔟) = g − 1 (𝔠). Recently, Kabbaj, Louartiti and Tamekkante defined and studied the following subring: A ⋈ f , g (𝔟 , 𝔠) : = { (f (a) + b , g (a) + c) | a ∈ A , b ∈ 𝔟 , c ∈ 𝔠 } of B × C , called the bi-amalgamation of A with (B , C) along (𝔟 , 𝔠) , with respect to (f , g). This ring construction is a natural generalization of the amalgamated algebras, introduced and studied by D'Anna, Finocchiaro and Fontana. The aim of this paper is to continue the investigation started by Kabbaj, Louartiti and Tamekkante by providing a deeper insight on the ideal-theoretic structure of bi-amalgamations. [ABSTRACT FROM AUTHOR]

The class of Pascal finite polynomial automorphisms is a subclass of the class of locally finite ones allowing a more effective approach. In characteristic zero, a Pascal finite automorphism is the exponential map of a locally nilpotent derivation. However, Pascal finite automorphisms are defined in any characteristic, and therefore constitute a generalization of exponential automorphisms to positive characteristic. In this paper, we prove several properties of Pascal finite automorphisms. We obtain in particular that the Pascal finite property is stable under taking powers but not under composition. This leads us to formulate a generalization of the exponential generators conjecture to arbitrary characteristic. [ABSTRACT FROM AUTHOR]

In this paper, firstly we study the central invariant ZH(L)0 of generalized Hom-Lie algebras L, which are monoidal Hom-Lie algebras in the category of H-modules for a triangular Hopf algebra (H, R). If V is an H-Hom-Lie ideal of [L, L], under certain conditions we obtain V0 ⊆ ZH(L)0, where V0 is the monoidal Hom-subalgebra of H-invariant of V. Secondly, we describe the H-Hom-Lie ideal structure of monoidal Hom-algebras in a module category. [ABSTRACT FROM AUTHOR]

A subgroup H of a finite group G is called an -subgroup in G if there exists a normal subgroup T of G such that G = HT and Hg ∩ NT(H) ≤ H for all g ∈ G. In this paper, we continue to investigate the structure of a finite group G with -subgroups. Some new results are given and many known results are generalized. [ABSTRACT FROM AUTHOR]

In this paper, we shall introduce thin n-subpolygroups of a given n-polygroup and in this regards, the notion of wreath product of n-polygroups will be studied. Also, double cosets of n-polygroups are investigated and the classical isomorphism theorems of groups are generalized to n-polygroups. The main result of the paper is that a finite n-polygroup is singular if and only if it is a wreath product of n-subpolygroups all of which are thin or generated by an involution or by an idempotent element. [ABSTRACT FROM AUTHOR]

It is well known that a polynomial f (x) over a commutative ring R with identity is a zero-divisor in R [ x ] if and only if f (x) has a non-zero annihilator in the base ring, where R [ x ] is the polynomial ring with indeterminate x over R. But this result fails in non-commutative rings and in the case of formal power series ring. In this paper, we consider the problem of determining some annihilator properties of the formal power series ring R [ [ x ] ] over an associative non-commutative ring R. We investigate relations between power series-wise McCoy property and other standard ring-theoretic properties. In this context, we consider right zip rings, right strongly A B rings and rings with right Property (A). We give a generalization (in the case of non-commutative ring) of a classical results related to the annihilator of formal power series rings over the commutative Noetherian rings. We also give a partial answer, in the case of formal power series ring, to the question posed in [1 Question, p. 16]. [ABSTRACT FROM AUTHOR]

In this paper, we study the property of the Arf good subsemigroups of ℕ n , with n ≥ 2. We give a way to compute all the Arf semigroups with a given collection of multiplicity branches. We also deal with the problem of determining the Arf closure of a set of vectors and of a good semigroup, extending the concept of characters of an Arf numerical semigroup to Arf good semigroups. [ABSTRACT FROM AUTHOR]

In [A. Hamed and S. Hizem, Modules satisfying the S -Neotherian property and S -ACCR, Comm. Algebra44 (2016) 1941–1951], the authors introduced the notion of S -stationary sequence, which is a generalization of the ascending chain condition (ACC) property. Let D be an integral domain and S a multiplicative subset of D. An increasing sequence (I k) k ∈ ℕ of ideals of D is called S -stationary if there exist a positive integer n and an s ∈ S such that for each k ≥ n , s I k ⊆ I n. As a generalization of Mori domains, we define D to be an S -Mori domain if every increasing sequence of integral divisorial ideals of D is S -stationary. In this paper, we investigate the class of such domains D and we give an S -version of classical results about Mori domains. [ABSTRACT FROM AUTHOR]

In this paper, we investigate commutativity of prime rings R with involution ∗ of the second kind in which generalized derivations satisfy certain algebraic identities. Some well-known results characterizing commutativity of prime rings have been generalized. Furthermore, we provide an example to show that the restriction imposed on the involution is not superfluous. [ABSTRACT FROM AUTHOR]

The covering number of a finite group G , denoted σ (G) , is the smallest positive integer k such that G is a union of k proper subgroups. We calculate σ (G) for a family of primitive groups G with a unique minimal normal subgroup N , isomorphic to A n m with n divisible by 6 and G / N cyclic. This is a generalization of a result of Swartz concerning the symmetric groups. We also prove an asymptotic result concerning pairwise generation. [ABSTRACT FROM AUTHOR]

We study the natural class of rings where each nilpotent element generates a nilpotent ideal, calling them the strongly 2-primal rings. We derive many basic properties of these rings, analyze their behavior under standard ring constructions and extensions, and taxonomize their relationship to other natural generalizations of commutativity. A slightly stronger condition is to assume that any nilpotent element generates a nilpotent ideal of the same index of nilpotence. We find that the difference between these two properties explains divergent behaviors in direct products, polynomial rings, Morita contexts, and other constructions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce type absolute direct summand (type-ADS) modules and rings as a natural generalization of ADS ones. Besides basic properties and characterizations of the notion, we present several examples illustrating borders of the theory. We also show that some particular classical classes of rings, such as commutative or right non-singular rings are type-ADS. [ABSTRACT FROM AUTHOR]