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AbstractFollowing a suggestion in a footnote of the paper [1] by Fano, we give a direct proof of the fact that one of the two Fano threefolds of degree 72 in P38 is isomorphic to the anticanonically embedded weighted projective space P(1,1,4,6). [ABSTRACT FROM AUTHOR]

This article is intended to indicate and discuss some errors in N. Lavi’s paper “A Ganzstellensatz for semi-algebraic sets and a boundedness criterion for rational functions”. [ABSTRACT FROM AUTHOR]

We analyze the proof of the main result of a paper by Cuadra, Etingof and Walton, which says that any action of a semisimple Hopf algebraHon thenth Weyl algebraA = An(K) over a fieldKof characteristic 0 factors through a group algebra. We verify that their methods can be used to show that any action of a semisimple Hopf algebraHon an iterated Ore extension of derivation typein characteristic zero factors through a group algebra. [ABSTRACT FROM AUTHOR]

In this paper we investigate commuting involution graphs in classical affine Weyl groups. Let W be a classical Weyl group of rank n, with W ˜ its corresponding affine Weyl group. Our main result is that if X is a conjugacy class of involutions in W ˜ , then the commuting involution graph C (W ˜ , X) is either disconnected or has diameter at most n + 2. This bound is known to hold for types A ˜ n and C ˜ n , so the main work of this paper is to prove the theorem for types B ˜ n and D ˜ n. [ABSTRACT FROM AUTHOR]

A theorem in the article “Characterization of Monoids by Properties of Generators,” written by Kilp and Knauer, states that amalgamated coproducts of generators are generators in the categoryAct-S of right S-acts over a monoid S. In this note we give a counterexample to deny this theorem. However, we show that the results in that article based on the mentioned theorem still remain valid. Moreover, an incorrect proposition in the monographMonoids, Acts and Categoriesby Kilp et al. which states that coamalgamated product of cogenerators are cogenerators is denied by a counterexample. [ABSTRACT FROM AUTHOR]

Let be a proper closed subset in the projective n-space over a field k of characteristic zero and let C(Y) be its affine cone. Let be the homogeneous defining ideal of Y with homogeneous maximal ideal. In this paper, we find out a lower bound of de Rham depth of C(Y) in terms of. [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]

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]

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]

This paper investigates the relation between the almost Gorenstein properties for graded rings and for local rings. Once R is an almost Gorenstein graded ring, the localization RM of R at the graded maximal ideal M is almost Gorenstein as a local ring. The converse does not hold true in general ([7, Theorems 2.7, 2.8], [8, Example 8.8]). However, it does for one-dimensional graded domains with mild conditions, which we clarify in the present paper. We explore the defining ideals of almost Gorenstein numerical semigroup rings as well. [ABSTRACT FROM AUTHOR]

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]

This very short correction notes a gap in an argument of an earlier paper, and also provides a theorem of similar flavor to the main result of that paper. [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]

Köthe's classical problem posed by G. Köthe in 1935 asks to describe the rings R such that every left R-module is a direct sum of cyclic modules (these rings are known as left Köthe rings). Köthe, Cohen and Kaplansky solved this problem for all commutative rings (that are Artinian principal ideal rings). During the years 1962 to 1965, Kawada solved Köthe's problem for basic finite-dimensional algebras. But, so far, Köthe's problem was open in the non-commutative setting. Recently, in the paper [Several characterizations of left Köthe rings, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. (2023) (to appear)]], we classified left Köthe rings into three classes one contained in the other: left Köthe rings, strongly left Köthe rings and very strongly left Köthe rings, and then, we solved Köthe's problem by giving several characterizations of these rings in terms of describing the indecomposable modules. In this paper, we will introduce the Morita duals of these notions as left co-Köthe rings, strongly left co-Köthe rings and very strongly left co-Köthe rings, and then, we give several structural characterizations for each of them. [ABSTRACT FROM AUTHOR]

For a graph G = (V , E) the edge ring k [ G ] is k [ x 1 , ... , x n ] / I (G) , where n = | V | and I(G) is generated by { x i x j ; { i , j } ∈ E } . The conjecture we treat is the following. Conjecture 1. If k [ G ] has a 2-linear resolution, then the projective dimension of K [ G ] , pd (k [ G ]) , equals the maximal degree of a vertex in G. As far as we know, this conjecture is first mentioned in a paper by Gitler and Valencia [7, Conjecture 4.13], and there it is called the Eliahou-Villarreal conjecture. The conjecture is treated in a recent paper by Ahmed, Mafi, and Namiq [1]. That there are counterexamples was noted already by Moradi and Kiani [9]. By interpreting k [ G ] as a Stanley-Reisner ring, we are able to characterize those graphs for which the conjecture holds. [ABSTRACT FROM AUTHOR]

A ring idempotent e ∈ R is said to be quarter-central (or q-central for short) if e R (1 − e) R e = 0 . If all idempotents in a ring R are q-central, R is said to be a quarter-abelian ring (or more simply, a q-abelian ring). In this paper, we show that such a ring R is characterized by the property that [ e , R ] [ e , R ] = 0 for all idempotents e ∈ R , where [ x , y ] denotes the additive commutator x y − y x . For any nonzero ring S and any integer n ≥ 3 , we show that the ring T n (S) of n × n upper triangular matrices over S is not q-abelian. On the other hand, T 2 (S) is q-abelian iff S is abelian (in the classical sense that all idempotents are central in S ). A final section of this paper relates q-central idempotents to the notion of exchange rings and the study of regular, unit-regular, and strongly regular elements in arbitrary rings. From the viewpoint of currently prevailing generalizations of abelian rings in the literature, q-abelian rings are situated between the class of semiabelian rings and the class of strongly IC rings. [ABSTRACT FROM AUTHOR]

In [2003], without giving a detailed proof, Yamauchi provided a formula to calculate the genus of a certain family of smooth complete intersection algebraic curves. That formula is used extensively by Beshaj, Shaska, and Shor to study the algebraic curves for which their Jacobian has superelliptic components. In this note, we determine the correct version of the genus formula with an algebraic proof. Then, we show that the Yamauchi's formula works only under certain conditions. [ABSTRACT FROM AUTHOR]

Let G be a finite group. Recently various functions are defined related to the set of order elements of G and using these functions, some interesting criteria for solvability, nilpotency, supersolvability and etc., are obtained. In this paper, we continue this work and for a finite group G, we consider the function m (G) = ∑ g ∈ G 1 / o (g) , where o(g) denotes the order of g ∈ G . In this paper, we prove that if m (G) < m (A 5) , m (G) < m (A 4) , m (G) < m (S 3) , m (G) < m (Q 8) or m (G) < m (C 2 × C 2) , then G is solvable, supersolvable, nilpotent, abelian or cyclic, respectively. [ABSTRACT FROM AUTHOR]

This work is devoted to the complete classification of solvable Leibniz algebras with the naturally graded quasi-filiform non-Lie nilradical. It should be noted that the class of naturally graded quasi-filiform Leibniz algebras is sufficiently large, and solvable Leibniz algebras with the nilradical which is isomorphic to several of these algebras are classified in a series of works. This paper considered all omitted cases and completed the problem of classifying solvable Leibniz algebras with naturally graded quasi-filiform non-Lie nilradical. At the end of the paper we combine all results obtained to date, and provide a complete list of solvable Leibniz algebras, whose nilradical is a naturally graded quasi-filiform Leibniz algebra. [ABSTRACT FROM AUTHOR]

Let o (G) be the average of the element orders of a finite group G. A research topic concerning this quantity is understanding the relation between o (G) and o (H) , where H is a subgroup of G. Let N be the class of finite nilpotent groups and let L (G) be the subgroup lattice of G. In this paper, we show that the set { o (G) o (H) | G ∈ N , H ∈ L (G) } is dense in [ 0 , ∞) . Other density results are outlined throughout the paper. [ABSTRACT FROM AUTHOR]

In this paper, we show that the cover ideals of theta graphs are normal square-free monomial ideals, and also satisfy the strong persistence property. Moreover, we re-prove a well-known result which is related to the cover ideals of bipartite graphs. We finally terminate this paper by stating some theorems which introduce some classes of normal monomial ideals. [ABSTRACT FROM AUTHOR]

In regard to two recent publications in the Mediterranean J. Math. (2021) and Forum Math. (2021) related to fully and characteristically inert socle-regularity, respectively, we define and study the so-called weakly characteristically inert socle-regular groups. In that aspect, as a culmination of the investigations of this sort, some more global results are obtained and, moreover, some new concrete results concerning the weakly fully inert socle-regular groups, defined as in the firstly mentioned above paper, are also established. In particular, we prove that all torsion-complete groups are characteristically inert socle-regular, which encompasses an achievement from the secondly mentioned paper and completely settles the problem posed there about this class of groups. [ABSTRACT FROM AUTHOR]

The Hochschild homology of the ring k [ x 1 , x 2 , ... , x d ] / (x 1 , x 2 , ... , x d) 2 has been known and calculated several ways. This paper uses those calculations to calculate cyclic, negative cyclic, and periodic cyclic homology of k [ x 1 , x 2 , ... , x d ] / (x 1 , x 2 , ... , x d) 2 over k for k = Z. These calculations have been known for k = Q or fields containing it, but this paper studies the torsion which one gets for k = Z and through it, via the universal coefficient theorem, for general k. The computations are related to homology, cohomology, and Tate cohomology of cyclic groups. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the description of local derivations on solvable Lie algebras of maximal rank. Namely, we consider a solvable Lie algebra of the form R = Q ⊕ N , where Q is the maximal torus subalgebra of R , N is the nilradical of R and dim Q = dim N / N 2. We prove that any local derivation of such solvable Lie algebra R is a derivation. Further, we present two examples of solvable Lie algebras which satisfy the condition dim Q < dim N / N 2 and the first algebra admit a local derivation which is not a derivation, while for the second algebra we prove that any local derivation is a derivation. We also apply the main result of the paper to the description of local derivations on so-called standard Borel subalgebras of complex simple Lie algebras. [ABSTRACT FROM AUTHOR]

In this paper we study the classes of superconnected and superfaithful left quasigroups, that are relevant in the study of Mal'cev varieties of left quasigroups. Then we focus on quandles and in particular to the involutory ones. We extend the main result of [a paper by Erofeeva] to the infinite case and we offer a characterization of several classes of involutory quandles in terms of the properties of the canonical generators of the displacement group, improving the main results of [a paper by Nobusawa]. [ABSTRACT FROM AUTHOR]

Let R be a noncommutative prime ring with involution ′ * ′ and let Q m s (R) be the maximal symmetric ring of quotients of R. In the present paper, we describe the structure of generalized Jordan *-derivations, i.e., additive mappings F : R → R satisfying F (x 2) = F (x) x * + x d (x) for all x ∈ R , where d is an associated Jordan *-derivation of R. Precisely, we prove that under certain conditions any generalized Jordan *-derivation of R is of the form F (x) = x a + b x * for all x ∈ R , where a , b ∈ Q m s (R). As a consequence, we show that any generalized Jordan *-derivation on a prime ring with involution ′ * ′ is generalized X-inner, provided R is not a PI-ring. Finally, we conclude our paper with a direction for further research. [ABSTRACT FROM AUTHOR]

In the present paper we study the Jordan counterparts of Rickart and Baer * -algebras, i.e., inner RJ-algebras and inner BJ-algebras and prove that a nilpotent Jordan algebra which has no square root nilpotent elements is an inner RJ-algebra. Also we explain that a nilpotent Jordan algebra that has no nilpotent elements with a square root b such that b 3 ≠ 0 is not an inner RJ-algebra if there exists a nonzero element a such that a 2 ≠ 0 . As a main result of the paper we give a description of a finite-dimensional inner RJ-algebra A , isomorphic to R + ̇ N , with a nilradical N and a finite-dimensional inner BJ-algebra with a nilradical N. [ABSTRACT FROM AUTHOR]

Given a chain of groups G 0 ≤ G 1 ≤ G 2 ... , we may form the corresponding chain of their representation rings, together with induction and restriction operators. Let Res l denote the operator which restricts down l steps, and similarly for Ind l. Observe that Ind l Res l is an operator from any particular representation ring to itself. We provide explicit rigid constraints that a group of chain with surjective restriction operators satisfying the polynomial property must obey. The central question that this paper addresses is: "What happens if the Ind l Res l operator is a polynomial in the Ind Res operator?" It is well known that chains of wreath products { H n ⋊ S n } n ∈ N have this property. In this paper, we deduce rigid numerical constraints any arbitrary chains of groups with surjective restriction operator and the aforementioned polynomial property must satisfy. These polynomials can be obtained from the orders of the groups, and are uniquely parametrized by two integers. We deduce the desired parametrization from deducing character theoretic properties of group chains with the desired polynomials. [ABSTRACT FROM AUTHOR]

Groups of structure 2. O 8 + (2) have an irreducible representation of degree 8 which can be realized over Z and any prime field F p. This representation extends to a group of structure 2. O 8 + (2).2. Any subgroup Z ≤ F p × acts by scalar multiplication on this module over F p. In this short note we determine for which primes p > 7 and which Z the central products Z ° (2. O 8 + (2) and Z ° (2. O 8 + (2).2) have a regular orbit on the 8-dimensional F p -module. This work was triggered by an omission in the paper [4] by Köhler and Pahlings, a paper which is used in various places in work on the k(GV)-problem. [ABSTRACT FROM AUTHOR]

In this paper, for any prime p, we propose the notion of a p-transitive association scheme. This notion aims to generalize the fact that the regular module of a group algebra of a finite group has a unique principal submodule to the case of the regular modules of modular adjacency algebras of association schemes. We characterize the p-transitive quasi-thin association schemes and the p-transitive association schemes with thin thin residue by their structure theory properties. We also get some results with independent interests. [ABSTRACT FROM AUTHOR]

For a fixed integer n≥2, let Mn be the algebra of all n × n matrices over the complex field C. Let x1,x2∈Cn be two fixed nonzero vectors, and fix also y1,y2∈Cn. In this paper, we characterize linear bijective maps φ:Mn→Mn having the property that φ(A)φ(B)x2=y2 whenever ABx1=y1. [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]

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 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]

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]

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]

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]

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]

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]

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 provide a new approach to construct monoidal Hom-Hopf algebras. We investigate monoidal Hom-Hopf algebra structure on a left (n, l)-Hom-crossed coproduct structure with a left n-Hom-smash product structure, obtaining Radford [ n , (n , l) ] -biproduct structure theorem. Then, we study a Hom-coaction admissible mapping system to characterize this Radford [ n , (n , l) ] -biproduct structure. Finally, we study the cosemisimplicity of a special Hom-smash coproduct and prove the related Maschke theorem. [ABSTRACT FROM AUTHOR]

In 1994, Long and Moody introduced a method to construct a new representation of the braid group from the representation of the braid group or the semidirect product of the braid group and the free group. In this paper, we show that its matrix presentation is written using the Fox derivative, and also a relation with twisted Alexander invariants. Communicated by Eric Jespers [ABSTRACT FROM AUTHOR]

Let Γ be a torsion-free cancellative monoid and let R = ⊕ α ∈ Γ R α be a graded integral domain. In this paper, we show that if R is a graded Strong Mori domain, then every homogeneously t-linked overring of R contained in R hwg is a graded Strong Mori domain, where Rhwg is the homogeneous w-global transform of R; and if R is a graded Strong Mori domain, then R S is also a graded Strong Mori domain, where S is a multiplicative set of ideals of R generated by homogeneous ideals of R. [ABSTRACT FROM AUTHOR]

Let G be a generalized matrix algebra. A linear map ϕ : G → G is said to be a left (right) Lie centralizer at E ∈ G if ϕ ([ S , T ]) = [ ϕ (S) , T ] ( ϕ ([ S , T ]) = [ S , ϕ (T) ] ) holds for all S , T ∈ G with ST = E. ϕ is of a standard form if ϕ (A) = Z A + γ (A) for all A ∈ G , where Z is in the center of G and γ is a linear map from G into its center vanishing on each commutator [ S , T ] whenever ST = E. In this paper, we give a complete characterization of ϕ. It is shown that, under some suitable assumptions on G , ϕ has a standard form. [ABSTRACT FROM AUTHOR]

Let F q denote the finite field of order q. In this paper, we investigate the permutation property of the quadrinomial f (x) = x + a 1 x q (q − 1) + 1 + a 2 x q + a 3 x 2 (q − 1) + 1 over F q 2 , where q is odd and a 1 , a 2 , a 3 ∈ F q * . More precisely, we obtain the necessary and sufficient conditions for f (x) to be a permutation polynomial over F q 2 . [ABSTRACT FROM AUTHOR]

In this paper, we study the class of affine semigroups generated by integral vectors, whose components are in generalized arithmetic progression and we observe that the defining ideal is determinantal. We also give a sufficient condition on the defining ideal of the semigroup ring for the equality of the Betti numbers of the defining ideal and those of its initial ideal. We introduce the notion of an affine semigroup generated by join of two affine semigroups and show that it preserves some nice properties, including Cohen-Macaulayness, when the constituent semigroups have those properties. [ABSTRACT FROM AUTHOR]