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In a finite group G, ρ (G) denotes the product of element orders of G. Let p , q , r , s and t be prime numbers. Recently, it was proved that C p 3 , Cpq, C p 2 q and Cpqrs are characterizable by the product of element orders, and if ρ (G) = ρ (C pqr) , then G ≅ A 5 or C pqr . In this paper, we continue this work and we show that C p × A 5 (for p > 5) and Cpqrst are characterizable by the product of element orders. In the rest of the paper, we show that if ρ (G) = ρ (C pqr) s ρ (C s) p 2 q , then (p , q , r , s) = (2 , 3 , 5 , 11) and G ≅ L 2 (11) . This shows that the structure of ρ (G) uniquely determined L 2 (11) and so L 2 (11) is the only group with this form of ρ (G) . [ABSTRACT FROM AUTHOR]

AbstractLet R be a commutative ring with identity, S⊆R be a multiplicative set, and M be an R-module. We say that a submodule N of M with (N:RM)∩S=∅ has an S-primary decomposition if it can be written as a finite intersection of S-primary submodules of M. In this paper, first we provide an example of the S-Noetherian module in which a submodule does not have a primary decomposition. Then our main aim of this paper is to establish the existence and uniqueness of S-primary decomposition in S-Noetherian modules as an extension of a classical Lasker-Noether primary decomposition theorem for Noetherian modules. [ABSTRACT FROM AUTHOR]

Finite commutative semifields of odd characteristic correspond to Dembowski-Ostrom polynomials. A new proof of this fact is the main result of this paper. The paper also discusses strong isotopy of commutative semifields and shows that the limit on the number of strong isotopy classes can be obtained from a general theorem on commutative loops. By this theorem for each commutative loop Q the commutative isotopes form at most | N μ : (N μ) 2 N λ | classes with respect to the strong isotopy, where N μ and N λ are the middle and the left nucleus of Q. Loops Q1 and Q2 are said to be strongly isotopic if there exists an isotopism Q 1 → Q 2 of the form (α , α , γ) . [ABSTRACT FROM AUTHOR]

AbstractThe notion of local rings with quasi-decomposable maximal ideal was formally introduced by Nasseh and Takahashi. In separate works, the authors of the present paper showed that such rings have rigid homological properties; for instance, they are both Ext- and Tor-friendly. One point of this paper is to further explore the homological properties of these rings and also introduce new classes of such rings from a combinatorial point of view. Another point is to investigate how far some of these homological properties can be pushed along certain diagrams of local ring homomorphisms. [ABSTRACT FROM AUTHOR]

AbstractIn the present paper corrected versions of the statements in the paper “Description of finite-dimensional inner Rickart and Baer Jordan algebras” by F.N. Arzikulov and U.I. Khakimov are given. In particular, it is shown that for any Jordan algebra J with an idempotent p and an associative degenerate radical D such that J=Fp+̇D, J is an inner RJ-algebra if and only if, for any nonzero a∈D, a2=0 and p(pa) = pa. Also, other equivalent conditions when a Jordan algebra J is an inner RJ-algebra are given. As for finite-dimensional nilpotent Jordan algebras, there is not a nilpotent inner RJ-algebra (and hence inner BJ-algebra) except the finite-dimensional Jordan algebra the square of each element of which is zero. [ABSTRACT FROM AUTHOR]

AbstractIn a previous paper we have compared the Hochschild cohomology groups of finite dimensional monomial algebras under gluing two idempotents. In the present paper, we compare the Hochschild cohomology groups of finite dimensional monomial algebras under gluing two arrows. [ABSTRACT FROM AUTHOR]

In a 1987 paper, Eliahou and Kervaire constructed a minimal resolution of a class of monomial ideals in a polynomial ring, called stable ideals. As a consequence of their construction they deduced several homological properties of stable ideals. Furthermore they showed that this resolution admits an associative, graded commutative product that satisfies the Leibniz rule. In this paper we show that their construction can be extended to stable ideals in skew polynomial rings. As a consequence we show that the homological properties of stable ideals proved by Eliahou and Kervaire hold also for stable ideals in skew polynomial rings. [ABSTRACT FROM AUTHOR]

In this paper, we progress in the general study of rational functional identities in the setting of division rings. The main theorem in the paper states that the only additive maps f and g satisfying that f (x) = − x n g (x − 1) , for n = 3 or 4, are the zero maps. In addition, we provide an alternative proof to the result proved by Dar and Jing regarding the classification of additive maps f and g satisfying that f (x) = − x 2 g (x − 1) . [ABSTRACT FROM AUTHOR]

In this paper, we define and study (co)homology theories of a compatible associative algebra. At first, we construct a new graded Lie algebra whose Maurer-Cartan elements are given by compatible associative structures on a vector space. Then we define the cohomology of a compatible associative algebra A and as applications, we study extensions, deformations and extensibility of finite order deformations of A. We end this paper by considering compatible presimplicial vector spaces and the homology of compatible associative algebras. [ABSTRACT FROM AUTHOR]

As a proper generalization of comultiplication modules, the concept of k-comultipli-cation semimodules is introduced in this paper. Let S be a semiring. An left S-semimodule M is called a left k-comultiplication semimodule if for each subtractive subsemimodule N of M, there exists an ideal I of S such that N = (0 : M r I) . Some properties of k-comultiplication semimodules are obtained and some conditions are given for a direct sum of k-comultiplication semimodules to be a k-comultiplication semimodule. [ABSTRACT FROM AUTHOR]

We show that any n-absorbing ideal must be strongly n-absorbing, which is the first of Anderson and Badawi's three interconnected conjectures on absorbing ideals. We prove this by introducing and studying objects called maximal and semimaximal collections, which are tools for analyzing the multiplicative ideal structure of commutative rings. Furthermore, we discuss and make heavy use of previous work on absorbing ideals done by Anderson and Badawi, Choi and Walker, Laradji, and Donadze, among others. At the end of this paper, we pose three conjectures, each of which implies the last unsolved conjecture made by Anderson and Badawi. [ABSTRACT FROM AUTHOR]

This paper aims to study the local derivations, 2-local automorphisms and local automorphisms on the super-Virasoro algebras. The primary focus is to establish that every local derivation of the super-Virasoro algebras is indeed a derivation, and to demonstrate that every local or 2-local automorphism of the super-Virasoro algebras is an automorphism. [ABSTRACT FROM AUTHOR]

Let k be an algebraically closed field of characteristic p ≥ 0 , let G be a simple simply connected classical linear algebraic group of rank l and let T be a maximal torus in G with rational character group X (T) . For a nonzero p-restricted dominant weight λ ∈ X (T) , let V be the associated irreducible kG-module. We define ν G (V) as the minimum codimension of any eigenspace on V for any non-central element of G. In this paper, we determine lower-bounds for ν G (V) for G of type A l and dim (V) ≤ l 3 2 , and for G of type B l , C l , or D l and dim (V) ≤ 4 l 3 . Moreover, we give the exact value of ν G (V) for G of type A l with l ≥ 15 ; for G of type B l or C l with l ≥ 14 ; and for G of type D l with l ≥ 16 . [ABSTRACT FROM AUTHOR]

For a commutative ring R and a Hopf algebra H which is finitely generated projective as an R-module, it is established that there is an (anti)-isomorphism of groups between the Brauer group BQ(R, H) of Hopf Yetter-Drinfel'd H-module algebras and the Brauer group BQ (R , H *) of Hopf Yetter-Drinfel'd H * -module algebras, where H * is the linear dual of H. In this paper, we generalize this result by establishing an anti-isomorphism of groups between BQ(S, H), the Brauer group of dyslectic Hopf Yetter-Drinfel'd (S, H)-module algebras and BQ (S o p , H *) , the Brauer group of dyslectic Hopf Yetter-Drinfel'd (S o p , H *) -module algebras, where S is an H-commutative Hopf Yetter-Drinfel'd H-module algebra and Sop is the opposite algebra of S. Communicated by Alberto Elduque [ABSTRACT FROM AUTHOR]

Let I (Γ) be the homogeneous ideal of a finite set Γ in P n of d points in linearly general position. In 1972, Saint-Donat proved that if d ≤ 2 n then I (Γ) is generated by completely decomposable quadratic polynomials. Later, R. Treger generalized this result by showing that I (Γ) is generated by forms of degree ≤ m if d ≤ mn for some m ≥ 2 . This paper aims to generalize Saint-Donat's work in a different direction by proving that the degree m piece of I (Γ) can be generated by completely decomposable forms of degree m if and only if d ≤ mn . Communicated by Daniel Erman [ABSTRACT FROM AUTHOR]

The present paper is devoted to study local derivations on the conformal Galilei algebra. We prove that every local derivation on the conformal Galilei algebra is a derivation. [ABSTRACT FROM AUTHOR]

Given a finite set A and a group homomorphism ϕ : H → G , a ϕ -cellular automaton is a function T : A G → A H that is continuous with respect to the prodiscrete topologies and ϕ -equivariant in the sense that h · T (x) = T (ϕ (h) · x) , for all x ∈ A G , h ∈ H , where · denotes the shift actions of G and H on A G and AH, respectively. When G = H and ϕ = id , the definition of id -cellular automata coincides with the classical definition of cellular automata. The purpose of this paper is to expand the theory of ϕ -cellular automata by focusing on the differences and similarities with their classical counterparts. After discussing some basic results, we introduce the following definition: a ϕ -cellular automaton T : A G → A H has the unique homomorphism property (UHP) if T is not ψ-equivariant for any group homomorphism ψ : H → G , ψ ≠ ϕ . We show that if the difference set Δ (ϕ , ψ) is infinite, then T is not ψ-equivariant; it follows that when G is torsion-free abelian, every non-constant T has the UHP. Furthermore, inspired by the theory of classical cellular automata, we study ϕ -cellular automata over quotient groups, as well as their restriction and induction to subgroups and supergroups, respectively. [ABSTRACT FROM AUTHOR]

In 1971, inspired by the work of Lazard and Govorov for R-modules over a ring R, B. Stenström proved that an S-act over a monoid S is isomorphic to a directed colimit of finitely generated free S-acts if and only if it is both pullback flat and equalizer flat. Such acts are now usually called strongly flat. In 1991, S. Bulman-Fleming discovered that every pullback flat S-act is in fact strongly flat by developing a new "interpolation" Condition (PF) for pullback-preservation. In 2005, V. Laan and S. Bulman-Fleming obtained a version of the Lazard-Govorov Theorem for S-posets over a pomonoid S, in which subpullbacks and subequalizers now take on the role previously played by pullbacks and equalizers. However, unlike the situation for acts, S. Bulman-Fleming showed in 2009 that subpullback flat S-posets need not be strongly flat, and also T. Zhao found in 2022 that Condition (PF) does not imply strongly flatness for S-posets. The present paper is devoted to the study of subpullback flatness and Condition (PF) in the context of S-posets over a pomonoid S. First we unexpectedly show that subpullback flatness is equivalent to Condition (PF). Furthermore, we give some classifications of pomonoids by subpullback flatness of (cyclic, Rees factor) S-posets, and give some classes of pomonoids that all (cyclic) S-posets have a PF-cover. Finally, we investigate products of subpullback flat S-posets. [ABSTRACT FROM AUTHOR]

In this paper, we will investigate the structure of groups with the property of the core relation between some second maximal subgroups and corresponding maximal subgroups, and obtained some new results about finite groups. ARTICLE HISTORY Received 7 September 2023 Revised 13 December 2023 Communicated by Mandi Schaeffer Fry [ABSTRACT FROM AUTHOR]

In the paper, we show that a uniformly bounded and nonnegative triangle functor between Gorenstein derived categories of two CM-algebras induces a Gorenstein-projectively stable functor between the corresponding Gorenstein-projectively stable categories. Furthermore, we show that a ladder of Gorenstein derived categories of CM-finite algebras induces a recollement of Gorenstein-projectively stable categories under certain nice conditions. As a byproduct, a Gorenstein derived equivalence induces a Gorenstein-projectively stable equivalence for CM-finite Gorenstein algebras. [ABSTRACT FROM AUTHOR]

Differential operators and integral operators are linked together by the first fundamental theorem of calculus. Based on this principle, the notion of a differential Rota-Baxter algebra was proposed by Guo and Keigher. Recently, the subject has attracted more attention since it is associated with many areas in mathematics, such as integro-differential algebras. This paper considers differential Rota-Baxter algebras in the quasi-idempotent operator context. We establish a Gröbner-Shirshov basis for free commutative quasi-idempotent differential algebras (resp. Rota-Baxter algebras, resp. differential Rota-Baxter algebras). This provides a linear basis of a free object in each of the three corresponding categories by the Composition-Diamond lemma. Communicated by P. Kolesnikov [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring and M be an R-module. The R-module M is called distributive if for every submodules S, T and U of M, the equality S ∩ (T + U) = S ∩ T + S ∩ U holds true. In this paper, we give a necessary and sufficient condition for M to be distributive based on injective envelopes. The proof uses Matlis' results on injective modules. [ABSTRACT FROM AUTHOR]

In this paper, we unfold balanced and totally balanced neighborhood hypergraphs to discover new classes of normally torsion-free monomial ideals. As a consequence, we establish that the closed neighborhood ideals and the dominating ideals of strongly chordal graphs are normally torsion-free. We discuss the stable sets of associated primes of the dominating ideals of cycles and characterize all the cycles with normally torsion-free dominating ideals. [ABSTRACT FROM AUTHOR]

In this paper we prove that every automorphism of a Chevalley group (or its elementary subgroup) with root system of rank > 1 over a commutative ring (with 1/2 for the systems A 2 , F 4 , B l , C l ; with 1/2 and 1/3 for the system G 2 ) is standard, i.e., it is a composition of ring, inner, central and graph automorphisms. This result finalizes description of automorphisms of Chevalley groups. However, the restrictions on invertible elements can be a topic of further considerations. We provide also some model-theoretic applications of this description. [ABSTRACT FROM AUTHOR]

In this paper, we show that the Δ-genus Δ (X , L) ≥ 0 for any connected polarized demi-normal scheme (X , L) over an algebraically closed field k. As an application, we obtain Δ (X , I (K X + Λ)) ≥ 0 for any KSBA stable log scheme (X , Λ) , where I is the Cartier index of K X + Λ . We also construct examples of KSBA stable log schemes with I = 1 and Δ (X , K X + Λ) = 0 , which shows the inequality is sharp when I = 1. [ABSTRACT FROM AUTHOR]

Let a be an ideal of Noetherian ring R and M a finitely generated R-module such that cd (a , M) = c . In this paper, we investigate Att R (H a c (M)) . Among other things, it is shown that Max { p ∈ Supp R M | cd (a , R / p) = c } ⊆ Att R (H a c (M)) . We also show that Att R (H a c (M)) = { p ∈ Supp R M | cd (a , R / p) = c , p = Ann R (H a c (R / p)) } and { p ∈ Supp R M | cd (a , R / p) = c , dim R / p − 1 ≤ cd (a , R / p) ≤ dim R / p } ⊆ Att R (H a c (M)). Finally, we prove that if (R , m) is a local ring and dim R / a = 1 then Att R (H a c (M)) = { p ∈ Supp R M | cd (a , R / p) = cd (a , M) } . Then by using this, it is shown that if (R , m) is a local ring then { p ∈ Supp R M | cd (a , R / p) = c , dim R / (a + p) = 1 } ⊆ Att R (H a c (M)). [ABSTRACT FROM AUTHOR]

There has been a great importance in understanding the nilpotent multipliers of finite groups in recent past. Let a group G be presented as the quotient of a free group F by a normal subgroup R. Given a positive integer c, the c-nilpotent multiplier of the group G is the abelian group M (c) (G) = (R ∩ γ c + 1 (F)) / γ c + 1 (R , F) . In particular, M (1) (G) is the Schur multiplier of G. The study of Schur multiplier of finite metacyclic groups goes back to the paper by F. R. Beyl in 1973. In this article, we study the 2-nilpotent multiplier of finite split metacyclic groups. [ABSTRACT FROM AUTHOR]

We study characterizations of sincere modules, sincere silting modules and tilting modules in terms of various vanishing conditions. Let R be a perfect ring and T be an R-module. It is proved that T is sincere silting if and only if T is presilting satisfing the vanishing condition KerExt R 0 ≤ i ≤ 1 (T , −) = 0 , and that T is tilting if and only if KerExt R 0 ⩽ i ⩽ 1 (T , −) = 0 and Gen T ⊆ KerExt R 1 ⩽ i ⩽ 2 (T , −) . As an application, we prove that a sincere silting R-module T of finite projective dimension is tilting if and only if Ext R i (T , T (J)) = 0 for all sets J and all integer i ≥ 1 . This not only extends a main result of Zhang's paper [Self-orthogonal τ-tilting modules and tilting modules, J Pure Appl Algebra, 2022, 226: 106860] from finitely generated modules over Artin algebras to infinitely generated modules over more general rings, but also gives it a different proof without using Auslander-Reiten translations. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we study modules having the property that are invariant under some idempotent endomorphisms of its injective envelope. Such modules are called essentially π-injective. It is shown that (1) M is essentially π-injective iff for any essentially finite direct summand X1 of M and any submodule X2 of M with X1∩X2=0 , there exists a direct summand X0 of M containing X2 such that M=X1⊕X0 , (2) M is essentially π-injective iff M is an ef-extending right R-module and for any decomposition M=M1⊕M2 with M1 essentially finite, M1 and M2 are relatively injective, (3) if M is essentially π-injective and R satisfies ACC on right ideals of the form r(m), m∈M , then M is a direct sum of uniform submodules. We also describe rings via essentially π-injective modules. It is shown that R is a semisimple artinian ring iff the direct sum of any two essentially π-injective right R-modules is essentially π-injective. [ABSTRACT FROM AUTHOR]

AbstractThe purpose of this paper is to explain a method on the generalization of the Bertini-type theorem on standard graded rings to the non-standard graded case of certain type. [ABSTRACT FROM AUTHOR]

AbstractIn this paper we introduce the notion of functional prime ideals in a commutative ring. For a (left) R-module M and a functional ϕ (i.e., an R-linear map ϕ from M to R), an ideal I of R is said to be a ϕ -prime ideal if whenever a∈R and m∈M such that aϕ(m)∈I , then a∈I or ϕ(m)∈I . This notion shows its ability to characterize different classes of ideals in terms of functional primeness with respect to specific R-modules. For instance, if the module M is the ideal I itself, then I is ϕ -prime for every ϕ∈HomR(I,R) if and only if I is a trace ideal, and if the module M is the dual of I, then I is ϕ -prime for every ϕ∈HomR(I−1,R) if and only if I is a prime ideal of R, or I is a strongly divisorial ideal. Several results are obtained and examples to illustrate the aims and scopes are provided. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we give a complete classification of torsion pairs in m-cluster categories of type D when m is odd, denoted by CDnm, via a bijection to combinatorial objects called Ptolemy diagrams of type D. As applications, we classify m-rigid subcategories of CDnm, which gives Jacquet-Malo’s classification of m-cluster tilting subcategories of CDnm. When m = 1, we generalizes the work of Holm, Jørgensen and Rubey [ 14] for the classification of torsion pairs in cluster categories of type Dn.Communicated by Steffen Oppermann [ABSTRACT FROM AUTHOR]

AbstractLet R be a commutative ring with identity and S a multiplicative subset of R. The aim of this paper is to study the class of commutative rings in which every S-flat module is flat (resp., projective). An R-module M is said to be S-flat if the localization of M at S, MS, is a flat RS-module. Commutative rings R for which all S-flat R-modules are flat are characterized by the fact that R/Rs is a von Neumann regular ring for every s∈S. While, commutative rings R for which all S-flat R-modules are projective are characterized by the following two conditions: R is perfect and the Jacobson radical J(R) of R is S-divisible. Rings satisfying these conditions are called S-perfect. Moreover, we give some examples to distinguish perfect rings, S-perfect rings, and semisimple rings. We also investigate the transfer results of the “S-perfectness” for various ring constructions, which allows the construction of more interesting examples. [ABSTRACT FROM AUTHOR]

AbstractLet M(S;Λ;P) denote a Rees I×Λ matrix semigroup over a semigroup S in which I is a singleton. In this paper we describe the right regular representation of M(S;Λ;P), and find connection between properties of S and M(S;Λ;P). We also prove embedding theorems on Rees matrix semigroups M(S;Λ;P). [ABSTRACT FROM AUTHOR]

AbstractLet (X,≤) be a locally finite pre-ordered set and R be a commutative ring with unity. In this paper we apply the theory of zero product determined algebras to show that each linear map on the incidence algebra I(X,R) which is derivable at zero is a generalized derivation and every local derivation on I(X,R) is a derivation. [ABSTRACT FROM AUTHOR]

AbstractBy means of the maximal left ring of quotients, this paper proves that every n-Lie m-derivation on a triangular ring can be resolved into a sum of an extremal n-derivation and an n-additive central mapping. As applications, n-Lie m-derivations on some classical triangular rings are characterized. [ABSTRACT FROM AUTHOR]

AbstractLet Γ be a torsion-free cancellative monoid and let R=⊕α∈ΓRα be a graded integral domain. In this paper, we define the concept of homogeneous star-operations, which is a generalization of that of star-operations, and apply some properties of star-operations. Also, we define the concept of homogeneous divisor classes, which is the graded integral domain version of that of divisor classes, and study some equivalent conditions for the set of homogeneous divisor classes HD(R) of R to be a group. [ABSTRACT FROM AUTHOR]

This paper deals with additive groups of rings in which all subrings are ideals. It is shown that if an abelian group supports only rings with this property, then all of them are commutative. This result is obtained for associative as well as not necessarily associative rings. Some previously known far from obvious results related to the mentioned groups are generalized and complemented. In particular, the classification of torsion-free abelian groups on which every not necessarily associative ring has the property that all its subrings are ideals is provided up to the structure of nil-groups. [ABSTRACT FROM AUTHOR]

The aim of this paper is to describe the irreducible modules for the Jordan algebra of 2 × 2 matrices over an algebraically closed field of characteristic different from 2, 3 and 5 in the class of the commutative power-associative algebras. All irreducible non-unital modules, and irreducible unital modules up to dimension three are classified, namely we find seven non-parameterized and five families of parameterized modules of dimension three. For every k ≥ 2 , an irreducible module of dimension 3k is also constructed. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a kind of decomposition of a finite group called a uniform group factorization, as a generalization of exact factorizations of a finite group. A group G is said to admit a uniform group factorization if there exist subgroups H 1 , H 2 , ... , H k such that G = H 1 H 2 ⋯ H k and the number of ways to represent any element g ∈ G as g = h 1 h 2 ⋯ h k ( h i ∈ H i ) does not depend on the choice of g. Moreover, a uniform group factorization consisting of cyclic subgroups is called a uniform cyclic group factorization. First, we show that any finite solvable group admits a uniform cyclic group factorization. Second, we show that whether all finite groups admit uniform cyclic group factorizations or not is equivalent to whether all finite simple groups admit uniform group factorizations or not. Lastly, we give some concrete examples of such factorizations. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to prove the positivity of double point divisors of general inner projections for projective S2-varieties with invertible dualizing sheaves, without assuming the smoothness or the normality of varieties. As applications, we characterize two types of projective varieties: varieties with trivial double point divisors and varieties with d − e − 1 ≤ n for dimension n, degree d and codimension e. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give some constructions results and examples of n-BiHom-Lie color algebras. Next, we introduce the definition of BiHom-modules over n-BiHom-Lie color algebras and we provide some properties. Moreover we investigate generalized derivations of n-BiHom-Lie color algebras and their BiHom-subalgebras. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring and Nil(R) denote the set of nilpotent elements of R. In this paper, we investigate two generalizations of exact sequences, ϕ -exact sequences and N -exact sequences. With the help of two generalizations of exact sequences, we generalize the concept of homology modules. We show that they behave in a way similar to the classical ones. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring. An R-module M is said to be w-split if Ext R 1 (M , N) is a GV-torsion R-module for all R-modules N. It is known that every projective module is w-split, but the converse is not true in general. In this paper, we study the w-split dimension of a flat module. To do so, we introduce and study the so-called w-copure (resp., strongly w-copure) projective modules, which is in some way a generalization of the notion of copure (resp., strongly copure) projective modules. An R-module M is said to be w-copure projective (resp., strongly w-copure projective) if Ext R 1 (M , N) (resp., Ext R n (M , N) ) is a GV-torsion R-module for all flat R-modules N and any n ≥ 1 . [ABSTRACT FROM AUTHOR]

Given a construction f on groups, we say that a group G is f -realisable if there is a group H such that G ≅ f (H) , and completely f-realisable if there is a group H such that G ≅ f (H) and every subgroup of G is isomorphic to f (H 1) for some subgroup H 1 of H and vice versa. Denote by L (G) the absolute center of a group G, that is the set of elements of G fixed by all automorphisms of G. By using the structure of the automorphism group of a ZM-group, in this paper we prove that cyclic groups C N , N ∈ N * , are completely L-realisable. [ABSTRACT FROM AUTHOR]

This paper describes properties of three certain classes of modules M over a ring R determined by conditions on isomorphic direct summands ( ≲ ⊕ ): The condition that whenever (I m λ ≅) M / Ker λ ≲ ⊕ M then Ker λ and I m λ are direct summands of M for any endomorphism λ ∈ End (M) (kernel-endoregular modules). The condition that if M / A ≅ B where A, B ≲ ⊕ M then M / B ≅ A (iso-summand-morphic modules). The condition if M / A ≅ B where A, B ≤ ⊕ M , then M / B ≅ A (summand-morphic modules) which is precisely the internal cancellation property for modules. [ABSTRACT FROM AUTHOR]

In this paper, we study the determinant of irreducible representations of the generalized symmetric groups Z r ≀ S n . We give an explicit formula to compute the determinant of an irreducible representation of Z r ≀ S n . Recently, several authors have characterized and counted the number of irreducible representations of a given finite group with nontrivial determinant. Motivated by these results, given an integer n, r an odd prime and ζ a nontrivial multiplicative character of Z r ≀ S n with n < r, we obtain an explicit formula to compute N ζ (n) , the number of irreducible representations of Z r ≀ S n whose determinant is ζ. [ABSTRACT FROM AUTHOR]

AbstractLet p be a prime and Fp be a finite field of p elements. Let FpG denote the group algebra of the finite p-group G over the field Fp and V(FpG) denote the group of normalized units in FpG. Suppose that G is a finite p-group given by a central extension of the form 1→Zpn×Zpm→G→Zp×⋯×Zp→1 and G′≅Zp, n,m≥1 and p is odd. In this paper, the structure of G is determined. And the relations of V(FpG)pl and Gpl, Ωl(V(FpG)) and Ωl(G) are given. Furthermore, there is a direct proof for V(FpG)p∩G=Gp. [ABSTRACT FROM AUTHOR]

AbstractLet a be an ideal of a given commutative Noetherian ring R which satisfies the condition of the first problem of R. Hartshorne in [Affine duality and cofiniteness, Invent. Math. 9 (1970), 145–164]. In this paper, we prove that a also satisfies the condition of his second problem in the same article. We also provide an example to show that the converse statement does not hold in general. [ABSTRACT FROM AUTHOR]

AbstractThe notion of double extension for symplectic Lie algebras was introduced by Medina and Revoy in 1985. In 2021, Valencia gave a condition for symplectic Lie algebras which are obtained through double extension to admit compatible left-symmetric structures. In this paper, we formulate the notion of double extension for left-symmetric structures and give a condition for left-symmetric structures to be obtained through double extension. Moreover, we prove that right nilpotent left-symmetric structures on some solvable Lie algebras are obtained through double extension. We also give a condition for left-symmetric structures which are obtained through double extension to be complete. [ABSTRACT FROM AUTHOR]