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Bera [Line graph characterization of power graphs of finite nilpotent groups, Comm. Algebra 50(11) (2022) 4652–4668] characterized finite nilpotent groups whose power graphs and proper power graphs are line graphs. In this paper, we extend the results of above-mentioned paper to arbitrary finite groups. Also, we correct the corresponding result of the proper power graphs of dihedral groups. Moreover, we classify all the finite groups whose enhanced power graphs are line graphs. We classify all the finite nilpotent groups (except non-abelian 2-groups) whose proper enhanced power graphs are line graphs of some graphs. Finally, we determine all the finite groups whose (proper) power graphs and (proper) enhanced power graphs are the complement of line graphs, respectively. [ABSTRACT FROM AUTHOR]

The SFT (for strong finite type) condition was introduced by [J. T. Arnold, Krull dimension in power series rings, Trans. Amer. Math. Soc. 177 (1973) 299–304] in the context of studying the condition for formal power series rings to have finite Krull dimension. In the context of commutative rings, the SFT property is a near-Noetherian property that is necessary for a ring of formal power series to have finite Krull dimension behavior. Many others have studied this condition in the context of the dimension of formal power series rings. In this paper, we explore a specialization (and in some sense a more natural) variant of the SFT property that we dub as the VSFT (for very strong finite type) property. As is true of the SFT property, the VSFT property is a property of an ideal that may be extended to a global property of a commutative ring with identity. Any ideal (respectively, ring) that has the VSFT property has the SFT property. In this paper, we explore the interplay of the SFT property and the VSFT property. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Temperley–Lieb algebras TLn(δ) at δ = 1. Since δ = 1, we can consider the multiplicative monoid structure and ask how this monoid acts on topological spaces. Given a monoid action on a topological space, we get an algebra action on each homology group. The main theorem of this paper explicitly deduces the representation structure of the homology groups in terms of a natural filtration associated with our TLn-space. As a corollary of this result, we are able to study stability phenomena. There is a natural way to define representation stability in the context of TLn(1), and the presence of filtrations enables us to define a notion of topological stability. We are able to deduce that a filtration-stable sequence of TLn-spaces results in representation-stable sequence of homology groups. This can be thought of as the analogue of the statement that the homology of configuration spaces forms a finitely generated FI-module. [ABSTRACT FROM AUTHOR]

The cyclic graph Γ (S) of a semigroup S is the simple undirected graph whose vertex set is S and two vertices x , y are adjacent if the subsemigroup generated by x and y is monogenic. In this paper, we determine the clique number of Γ (S) for an arbitrary semigroup S. Further, we obtain the independence number of Γ (S) if S is a finite monogenic semigroup. At the final part of this paper, we give bounds for independence number of Γ (S) if S is a semigroup of bounded exponent and we also characterize the semigroups attaining the bounds. [ABSTRACT FROM AUTHOR]

Over the years questions have arisen about T-ideals of (noncommutative) polynomials. But when evaluating a noncentral polynomial in subalgebras of matrices, one often has little control in determining the specific evaluations of the polynomial. One way of overcoming this difficulty in characteristic 0, is to reduce to multilinear polynomials and to utilize the representation theory of the symmetric group. But this technique is unavailable in characteristic p > 0. An alternative method, which succeeds, is the process of "hiking" a polynomial, in which one specializes its indeterminates in several stages, to obtain a polynomial in which Capelli polynomial is embedded, in order to get control on its evaluations. This method was utilized on homogeneous polynomials in the proof of Specht's conjecture for affine algebras over fields of positive characteristic. In this paper, we develop hiking further to nonhomogeneous polynomials, to apply to the "representability question." Kemer proved in 1988 that every affine relatively free PI algebra over an infinite field, is representable. In 2010, the first author of this paper proved more generally that every affine relatively free PI algebra over any commutative Noetherian unital ring is representable [A. Belov, Local finite basis property and local representability of varieties of associative rings, Izv. Russian Acad. Sci. (1) (2010) 3–134. English Translation Izv. Math. 74(1) (2010) 1–126]. We present a different, complete, proof, based on hiking nonhomogeneous polynomials, over finite fields. We then obtain the full result over a Noetherian commutative ring, using Noetherian induction on T-ideals. The bulk of the proof is for the case of a base field of positive characteristic. Here, whereas the usage of hiking is more direct than in proving Specht's conjecture, one must consider nonhomogeneous polynomials when the base ring is finite, which entails certain difficulties to be overcome. In Appendix A, we show how hiking can be adapted to prove the involutory versions, as well as various graded and nonassociative theorems. [ABSTRACT FROM AUTHOR]

In [V. I. Arnold, Simple singularities of curves, Proc. Steklov Inst. Math. 226(3) (1999) 20–28, Sec. 5, p. 32], Arnold writes: 'Classification of singularities of curves can be interpreted in dual terms as a description of "co-artin" subalgebras of finite co-dimension in the algebra of formal series in a single variable (up to isomorphism of the algebra of formal series)'. In the paper, such a description is obtained but up to isomorphism of algebraic curves (i.e. this description is finer). Let K be an algebraically closed field of arbitrary characteristic. The aim of the paper is to give a classification (up to isomorphism) of the set of subalgebras of the polynomial algebra K [ x ] that contains the ideal x m K [ x ] for some m ≥ 1. It is proven that the set = ∐ m , Γ (m , Γ) is a disjoint union of affine algebraic varieties (where Γ ∐ { 0 , m , m + 1 , ... } is the semigroup of the singularity and m − 1 is the Frobenius number). It is proven that each set (m , Γ) is an affine algebraic variety and explicit generators and defining relations are given for the algebra of regular functions on (m , Γ). An isomorphism criterion is given for the algebras in . For each algebra A ∈ (m , Γ) , explicit sets of generators and defining relations are given and the automorphism group Aut K (A) is explicitly described. The automorphism group of the algebra A is finite if and only if the algebra A is not isomorphic to a monomial algebra, and in this case | Aut K (A) | < dim K (A / A) where A is the conductor of A. The set of orders of the automorphism groups of the algebras in (m , Γ) is explicitly described. [ABSTRACT FROM AUTHOR]

The enhanced power graph of a finite group G is the simple undirected graph whose vertex set is G and two distinct vertices x,y are adjacent if x,y ∈〈z〉 for some z ∈ G. An L(2, 1)-labeling of graph Γ is an integer labeling of V (Γ) such that adjacent vertices have labels that differ by at least 2 and vertices distance 2 apart have labels that differ by at least 1. The λ-number of Γ, denoted by λ(Γ), is the minimum range over all L(2, 1)-labelings. In this paper, we study the lambda number of the enhanced power graph 풫E(G) of the group G. This paper extends the corresponding results, obtained in [X. Ma, M. Feng and K. Wang, Lambda number of the power graph of a finite group, J. Algebraic Combin. 53(3) (2021) 743–754], of the lambda number of power graphs to enhanced power graphs. Moreover, for a nontrivial simple group G of order n, we prove that λ(풫E(G)) = n if and only if G is not a cyclic group of order n ≥ 3. Finally, we determine the lambda number of the enhanced power graphs of nilpotent groups. [ABSTRACT FROM AUTHOR]

Recently, symplectic totally isotropic subspace inclusion graph ℐ n () on a 2 ν -dimensional symplectic space was introduced in [J. He, S. Zhang and G. Zhang, Symplectic totally isotropic subspace inclusion graph, Linear Multilinear Algebra (2022), doi:10.1080/03081087.2022.2112016], which is a graph whose vertices are all totally isotropic subspaces of and two distinct vertices are adjacent if and only if one is contained in the other. In that paper, the authors studied the diameter, girth, clique number and chromatic number of ℐ n () and also studied some other properties of ℐ n (). In this paper, we determine the automorphism group of ℐ n (). [ABSTRACT FROM AUTHOR]

A subset S of an integral domain is called a semidomain if the pairs (S, +) and (S∖{0},⋅) are commutative and cancellative semigroups with identities. The multiplication of S extends to the group of differences 풢(S), turning 풢(S) into an integral domain. In this paper, we study the arithmetic of semisubtractive semidomains (i.e. semidomains S for which either s ∈ S or − s ∈ S for every s ∈풢(S)). Specifically, we provide necessary and sufficient conditions for a semisubtractive semidomain to be atomic, to satisfy the ascending chain condition on principals ideals, to be a bounded factorization semidomain, and to be a finite factorization semidomain, which are subsequent relaxations of the property of having unique factorizations. In addition, we present a characterization of factorial and half-factorial semisubtractive semidomains. Throughout the paper, we present examples to provide insight into the arithmetic aspects of semisubtractive semidomains. [ABSTRACT FROM AUTHOR]

The authors of A. Benhissi and S. Hizem [Nonnil-Noetherian rings and SFT property, Rocky Mountain J. Math. 41 (2011) 1483–1500] proved that the primary decomposition of a nonnil-ideal of a nonnil-Noetherian ring always exists. We will show in this paper that this primary decomposition is minimal. We next prove that over any nonnil-Noetherian ring, a sub-class of modules admit a primary decomposition. In the second part of this paper, we give some properties related to ϕ-chained and ϕ-Prüfer rings. [ABSTRACT FROM AUTHOR]

Let R be an arbitrary ring equipped with an automorphism α and an α-derivation δ. In this paper, we study on the relationship between the piecewise prime property of a ring R and these of the skew inverse Laurent series ring R((x−1; α,δ)) and related rings. Furthermore, for a piecewise prime ring, we determine a large class of rings which have a generalized triangular matrix representation for which the diagonal rings are prime. Our results extend and unify many available results. Examples for illustration and delimitation of the theory are provided. [ABSTRACT FROM AUTHOR]

An integral domain R is perinormal if every local going-down overring is a localization of R and globally perinormal if every going-down overring is a localization of R. In this paper, I introduce notions of graded perinormal and graded globally perinormal domains and show that many results obtained for perinormal and globally perinormal domains have graded analogs. I also give some results for descent of properties between a graded domain and its 0th graded component. [ABSTRACT FROM AUTHOR]

Multiplicative differential (and the corresponding c-differential uniformity) was introduced by Ellingsen et al. in [C-differentials, multiplicative uniformity and (almost) perfect c-nonlinearity, IEEE Trans. Inf. Theory 66(9) (2020) 5781–5789], which has attracted lots of attention. Functions with low c-differential uniformity over finite fields, especially the PcN and APcN functions, have been widely investigated due to their applications in cryptography. In this paper, we first compute the c-differential uniformity of two classes of permutation polynomials. For one of these, we explicitly determine the c-DDT entries. For the second type of function, we give bounds for its c-differential uniformity. Besides, several classes of PcN or APcN functions are presented by employing some known functions and the (generalized) AGW criterion. [ABSTRACT FROM AUTHOR]

This paper describes the primary decomposition of the edge ideal of the join of weighted oriented graphs in terms of the information from each of those graphs. Using this decomposition and induction, we derive formulas for the depth and regularity of ordinary powers of the edge ideal of the join of two graphs consisting of isolated vertices. Furthermore, we provide upper bounds for the regularity of symbolic powers of such an edge ideal. For the edge ideal of the join of two graphs with at least one oriented edge for each graph, we provide exact formulas for their depth and regularity, as well as upper bounds for the regularity of ordinary powers of such an edge ideal. Some examples demonstrate that these upper bounds are sometimes attainable and sometimes strict. [ABSTRACT FROM AUTHOR]

Let G be a finite nilpotent group. We denote by γ(G) the number of conjugacy classes of noncyclic subgroups of G, Con(G) the number of conjugacy classes of subgroups of G. In this paper, we give the formula for γ(G), and prove that Con(G) − γ(G) ≥ 4 when the order of G is prime power. Moreover, we get the minimum of γ(G) according to t, where t is the number of distinct noncyclic Sylow subgroups of G. [ABSTRACT FROM AUTHOR]

In this paper, we study Betti splittings of cover ideals of bipartite graphs. We prove that if J ⊂ 한[x1,…,xn] is the cover ideal of a bipartite graph then the xi-partition of J is a Betti splitting for any i. We also prove that multigraded Betti numbers of any squarefree monomial ideal can appear in a certain part of multigraded Betti numbers of the cover ideal of a bipartite graph. [ABSTRACT FROM AUTHOR]

As it is known, the defining identities of a free Novikov algebra can be obtained from a commutative algebra with a derivation. In this paper, we consider a class of algebras obtained from the class of associative algebras with a derivation that generalizes Novikov algebras. Such objects are called noncommutative Novikov algebras. We construct a monomial basis for a free noncommutative Novikov algebra. [ABSTRACT FROM AUTHOR]

Let R be a polynomial ring in n indeterminates with coefficients in a field K of characteristic p > 0, and 풟 be the ring of differential operators over R. We consider 픭 as a nonmaximal homogeneous prime ideal, and ∗E(R/픭) as a graded injective hull of R/픭. In this paper, we prove the generalized Eulerian property for the graded injective hull R/픭 as a 풟-module in positive characteristic. Furthermore, we show that the minimal injective resolution of F-finite module ends in an object which is the direct sum of indecomposable in graded case using some finiteness properties in prime characteristic. [ABSTRACT FROM AUTHOR]

In this paper, we define a window code to be the portion of a Spatially-coupled low-density parity check (SC-LDPC) code seen by a single iteration of a windowed decoder. We consider the design of SC-LDPC codes for windowed decoding via optimization of the window code. In particular, because iterative decoding is optimal on codes with cycle-free graph representations, we ask fundamental questions about the construction and parameters of cycle-free window codes. We show that it is possible to have an SC-LDPC code with cycles and with cycle-free window codes. We consider the relationship between the distance of the window code and the distance of the SC-LDPC code. Further, we show that SC-LDPC codes with MDS window codes exist, and all such codes are asymptotically bad. This work gives insight into the tradeoffs between window code parameters and performance of the SC-LDPC code. [ABSTRACT FROM AUTHOR]

There is a non-unital ring I of order 4 defined by generators and relations as I = 〈 a , b | 2 a = 2 b = 0 , a 2 = b , a b = 0 〉. In this paper, we present special constructions of linear codes over I from the adjacency matrices of two class association schemes. These consist of either Strongly Regular Graphs (SRGs) or Doubly Regular Tournaments (DRTs). We investigate the conditions under which these codes are self-orthogonal, quasi self-dual, or Type IV. As a byproduct of this study, some diophantine relations on the weight of sum of vectors with coefficients in I are derived. Some examples of codes with minimum distance better than that of Type IV codes over unital rings of the same order in modest lengths are given. [ABSTRACT FROM AUTHOR]

The use of codes defined by sparse characteristic matrices, like QC-LDPC and QC-MDPC codes, has become an established solution to design secure and efficient code-based public-key encryption schemes, as also witnessed by the ongoing NIST post-quantum cryptography standardization process. However, similar approaches have been less fortunate in the context of code-based digital signatures, since no secure and efficient signature scheme based on these codes is available to date. The main limitation of previous attempts in this line of research has been the use of sparse signatures, which produces some leakage of information about the private key. In this paper, we propose a new code-based digital signature scheme that overcomes such a problem by publishing signatures that are abnormally dense, rather than sparse. This eliminates the possibility of deducing information from the sparsity of signatures, and follows a recent trend in code-based cryptography exploiting the hardness of the decoding problem for large-weight vectors, instead of its classical version based on small-weight vectors. We focus on one-time use and provide a first parameterization of the new scheme, the main advantages of which are very fast signature generation and verification, and public keys that are not too large for a code-based signature scheme. [ABSTRACT FROM AUTHOR]

In [D. R. L. Brown, Plactic key agreement (insecure?), J. Math. Cryptol.17(1) (2023) 20220010], a novel cryptographic key exchange technique was proposed using the plactic monoid, based on the apparent difficulty of solving division problems in that monoid. Specifically, given elements c , b in the plactic monoid, the problem is to find q for which q b = c , given that such a q exists. In this paper, we introduce a metric on the plactic monoid and use it to give a probabilistic algorithm for solving that problem which is fast for parameter values in the range of interest. [ABSTRACT FROM AUTHOR]

The construction of (minimal) linear codes from functions has received much attention in the literature. In this paper, we derive several minimal codes from the sets of pre-images of weakly regular plateaued and bent functions over the odd characteristic finite fields. Based on the recent construction method, we obtain six-weight and seven-weight minimal codes from these functions. The Hamming weights in proposed codes are calculated from the character sums of the sets based on the Walsh spectrum of the employed functions, while the weight distributions of the codes are determined from both the sizes of the employed sets and the Walsh distributions of the employed functions. [ABSTRACT FROM AUTHOR]

In this work, we analyze the problem of catastrophicity of encoders of convolutional codes over the Laurent series with coefficients in ℤ p r , ℤ p r ((d)). Kuijper and Pinto proved in [M. Kuijper and R. Pinto, On minimality of convolutional ring encoders, IEEE Trans. Autom. Control55(11) (2009) 4890–4897] that, contrary to what happens for codes over ((d)) , where is a field, when dealing with ℤ p r ((d)) there are convolutional codes that do not admit non-catastrophic encoders. Nevertheless it was conjectured that any catastrophic convolutional code admits another type of non-catastrophic encoder called p -encoder. In this paper we solve this conjecture for a class of (2 , 1) convolutional codes over ℤ p 2 and show that, in fact, these codes always admit a non-catastrophic p-encoder. We also describe a constructive procedure that allows us to obtain a non-catastrophic p -encoder. [ABSTRACT FROM AUTHOR]

In this paper, we study the relation between the linear subspace of the pseudo-noise (PN)-sequences generated by a primitive polynomial and the simplex code. This family of sequences can be also seen as an Maximum Distance Separable (MDS) 2 -linear code over 2 r . Furthermore, we see how to compute the family of generalized sequences produced by a primitive polynomial by means of a first-order Reed–Muller code. [ABSTRACT FROM AUTHOR]

In this paper, we study linear codes invariant under a cyclic endomorphism T , called T -cyclic codes. Since every cyclic endomorphism can be represented by a cyclic matrix with respect to a given basis, all of these matrices are similar. For simplicity we restrict ourselves to linear codes invariant under the right multiplication by a cyclic matrix M , that we call M -cyclic codes, and when M is the companion matrix C f of a given nonzero polynomial f (x) we call them f -cyclic codes. The similarity relation between matrices helps us find connections between M -cyclic codes and f -cyclic codes, where f (x) is the minimal polynomial of M. The class of M -cyclic codes contains cyclic codes and their various generalizations such as constacyclic codes, right and left polycyclic codes, monomial codes, and others. As common in the study of cyclic codes and their generalizations, we make use of the one-to-one correspondence between M -cyclic codes and ideals of the polynomial ring R f : = q [ x ] / 〈 f (x) 〉 , where f (x) is the minimal polynomial of M. This correspondence leads to some basic characterizations of these codes such as generator and parity check polynomials among others. Next, we study the duality of these codes, where we show that the b -dual of an M -cyclic code is an M ∗ -cyclic code, where M ∗ is the adjoint matrix of M with respect to b , and we explore some important results on the duality of these codes. Finally, we give examples as applications of some of the results and we construct some optimal codes. [ABSTRACT FROM AUTHOR]

In this paper, we give constructions for infinite sequences of finite nonlinear locally recoverable codes ⊆ ∏ i = 1 N q i over a product of finite fields arising from basis expansions in algebraic number fields. The codes in our sequences have increasing length and size, constant rate, fixed locality, and minimum distance going to infinity. [ABSTRACT FROM AUTHOR]

Maximum distance separable convolutional codes are characterized by the property that the free distance reaches the generalized Singleton bound, which makes them optimal for error correction. However, the existing constructions of such codes are available over fields of large size. In this paper, we present the unique construction of MDS convolutional codes of rate 1 / 2 and degree 5 over the field 1 1 . [ABSTRACT FROM AUTHOR]

Assume that R = ⊕n∈ℕ0Rn is a standard graded algebra over the local ring (R0, 픪0), 픞 is a homogeneous ideal of R, M is a finitely generated graded R-module and R+ := ⊕n∈ℕRn denotes the irrelevant ideal of R. In this paper, among other things, we study the asymptotic behavior of the set {grade(픞 ∩ R0,HR+grade(R+,M)(M) n)}n∈ℤ as n →−∞, in the case where 픞 and R+ are homogenously linked over M. [ABSTRACT FROM AUTHOR]

Let G be a graph with vertex set V (G), f a permutation of V (G). Define δf(x,y) = |d(x,y) − d(f(x),f(y))| and δf(G) =∑ δf(x,y), where the sum is taken over all unordered pairs x, y of distinct vertices of G. Let π(G) denote the smallest positive value of δf(G) among all permutations f of V (G). A permutation f with δf(G) = π(G) is called a near automorphism of G. In this paper, the near automorphisms of the complement or the square of a cycle are characterized. Moreover, π(Cn¯) and π Cn2 are determined. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to compute the nonzero semigroup determinant of the class of finite semigroups in which every two idempotents commute. This class strictly contains the class of finite semigroups that have central idempotents and the class of finite inverse semigroups. This computation holds significance in the context of the extension of the MacWilliams theorem for codes over semigroup algebras. [ABSTRACT FROM AUTHOR]

Let 풜0 be the class of Abelian block-rigid almost completely decomposable groups of ring type with cyclic regulator quotient. In this paper, we study relationships between the above groups G and their multiplication groups Mult G. It is proved that groups from 풜0 are determined by their multiplication groups. For a rigid group G ∈풜0, the isomorphism problem is solved: we describe multiplications from MultG that define isomorphic rings on G. We also describe Abelian groups that are realized as the multiplication group of some group in 풜0. We describe groups in 풜0 that are isomorphic to their multiplication groups. [ABSTRACT FROM AUTHOR]

We study the primeness of noncommutative polynomials on prime rings. Let R be a prime ring with extended centroid C, ρ a right ideal of R, f(X1,…,Xt) a noncommutative polynomial over C, which is not a polynomial identity (PI) for ρ, and a,b ∈ R∖{0}. Then af(x1,…,xt)b = 0 for all x1,…,xt ∈ ρ if and only if one of the following holds: (i) aρ = 0; (ii) ρC = eRC for some idempotent e ∈ RC and b ∈ ρC such that either f(X1,…,Xt)Xt+1 is a PI for ρ or f(X1,…,Xt) is central-valued on eRCe and ab = 0. We then apply the result to higher commutators of right ideals. Some results of the paper are also studied from the view of point of the notion of X-primeness of rings. [ABSTRACT FROM AUTHOR]

Permutation polynomials with low boomerang uniformity have wide applications in cryptography. In this paper, by utilizing the Weil sums technique and solving some certain equations over 픽2n, we determine the boomerang uniformity of these permutation polynomials: (1) f1(x) = (x2m + x + δ)22m+1 + x, where n = 3m, δ ∈ 픽2n with Trmn(δ) = 1; (2) f2(x) = (x2m + x + δ)22m−1+2m−1 + x, where n = 3m, δ ∈ 픽2n with Trmn(δ) = 0; (3) f3(x) = (x2m + x + δ)23m−1+2m−1 + x, where n = 3m, δ ∈ 픽2n with Trmn(δ) = 0. The results show that the boomerang uniformity of f1(x), f2(x) and f3(x) can attain 2n. [ABSTRACT FROM AUTHOR]

Let m be a cube-free positive integer and let p be a prime such that p ∤ m. In this paper, we find the number of conjugacy classes of completely reducible solvable cube-free subgroups in GL(2,q) of order m, where q is a power of p. [ABSTRACT FROM AUTHOR]

The small Davenport constant d(G) of a finite group G is defined to be the maximal length of a sequence over G which has no nontrivial product-one subsequence. In this paper, we prove that d(G) = 6 for the non-abelian group of order 27 and exponent 3 and thereby establish a conjecture by Gao and Zhuang for this group. [ABSTRACT FROM AUTHOR]

Let R be a ring with identity. The unit (respectively, unitary Cayley) graph of R is a simple graph G(R) (respectively, Γ(R)) with vertex set R, where two distinct vertices x and y are adjacent if and only if x + y (respectively, x − y) is a unit of R. In this paper, we explore when G(R) and Γ(R) are isomorphic for a finite ring R. Among other results, we prove that G(R) is isomorphic to Γ(R) for a finite ring R if and only if char(R/J(R))=2 or R/J(R)≅ℤ2 × S, where J(R) is the Jacobson radical of R and S is a finite ring. [ABSTRACT FROM AUTHOR]

Let R be a ring, σ be an automorphism of R and δ be a σ-derivation of R. We use Ω ⊆End(R, +) to denote the set of all words composed of σ, σ−1 and δ. A (σ,σ−1,δ)-ideal P of R is Ω-prime if whenever a,b ∈ R are such that aRω(b) ⊆ P for any ω ∈ Ω, we have a ∈ P or b ∈ P. In this paper, we first introduce the Ω-prime ideal and the Ω-prime radical of a ring R, to obtain connections between the prime radical of the Ore extension R[x; σ,δ] and the Ω-prime radical of the base ring R. Based on these results, we next give definitions of the Ω-LS-prime ideal, the Ω-strongly prime ideal and the Ω-uniformly strongly prime ideal of a ring R to provide formulas for the LS-prime radical, the strongly prime radical and the uniformly strongly prime radical of the Ore extension. [ABSTRACT FROM AUTHOR]

Let K be a field of characteristic zero and x a free variable. A K - ℰ -derivation of K [ x ] is a K -linear map of the form I , − , ϕ for some K -algebra endomorphism ϕ of K [ x ] , where I denotes the identity map of K [ x ]. In this paper, we study the image of an ideal of K [ x ] under some K -derivations and K - ℰ -derivations of K [ x ]. We show that the LFED conjecture proposed in [W. Zhao, Some open problems on locally finite or locally nilpotent derivations and ℰ -derivations, Commun. Contem. Math. 20(4) (2018) 1750056] holds for all K - ℰ -derivations and all locally finite K -derivations of K [ x ]. We also show that the LNED conjecture proposed in [W. Zhao, Some open problems on locally finite or locally nilpotent derivations and ℰ -derivations, Commun. Contem. Math. 20(4) (2018) 1750056] holds for all locally nilpotent K -derivations of K [ x ] , and also for all locally nilpotent K - ℰ -derivations of K [ x ] and the ideals u K [ x ] such that either u = 0 , or deg u ≤ 1 , or u has at least one repeated root in the algebraic closure of K. As a bi-product, the homogeneous Mathieu subspaces (Mathieu–Zhao spaces) of the univariate polynomial algebra over an arbitrary field have also been classified. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with zero-divisors Z (R) and i be a positive integer. The i -extended zero-divisor graph of R , denoted by Γ ¯ i (R) , is the (simple) graph with vertex set Z (R) ∗ = Z (R) \ { 0 } , the set of nonzero zero-divisors of R , and two distinct nonzero zero-divisors x and y are adjacent whenever there exist two positive integers n , m ≤ i such that x n y m = 0 with x n ≠ 0 and y m ≠ 0. The i -extended zero-divisor graph of R is well studied in 10. In this paper, we characterize the i -extended zero-divisor graphs of idealizations R ⋉ M (where M is an R -module). Namely, we study in detail the behavior of the filtration ( Γ ¯ i (R ⋉ M)) i ∈ ℕ ∗ as well as the relations between its terms. We also characterize the girth and the diameter of Γ ¯ i (R ⋉ M) and we give answers to several interesting and natural questions that arise in this context. [ABSTRACT FROM AUTHOR]

For an ample groupoid and a unit x of , Steinberg constructed the induction and restriction functors between the category of modules over the Steinberg algebra A R () and the category of modules over the isotropy group algebra R x . In this paper, we prove a graded version of these functors and give a description of spectral graded simple modules over the graded Steinberg algebra A R (). As an application, the spectral simple and graded simple modules over the Leavitt path algebra L K (E) are classified. In particular, we show that many of previously known simple and graded simple L K (E) -modules, including the Chen simple modules, are induced from (ungraded or graded) simple modules over isotropy group algebras. [ABSTRACT FROM AUTHOR]

Let G be a finite group and let χ be an irreducible character of G. If G / ker χ has a unique minimal normal subgroup, then χ is called monolithic. The character χ is said to be imprimitive if χ is induced from a character of a proper subgroup of G. In this paper, we classify the finite solvable groups having exactly two imprimitive monolithic characters whose kernels intersect trivially. [ABSTRACT FROM AUTHOR]

The paper contains the classification of non-commutative torsion-free associative rings of rank two. Furthermore, torsion-free abelian groups of rank two supporting an associative, but not commutative ring structure are classified. [ABSTRACT FROM AUTHOR]

This paper deals with some results concerning finitely generated coreduced comultiplication modules over a commutative ring. [ABSTRACT FROM AUTHOR]

For a character χ of a finite group G , the number cod (χ) = [ G : Ker χ ] χ (1) is called the co-degree of χ. In this paper, we concentrate on some conditions on the character co-degree sets implying solvability. More precisely, we show that if for every non-principal irreducible characters χ and ϕ of G with cod (χ) ≠ cod (ϕ) , the greatest common divisor of cod (χ) and cod (ϕ) is divisible by at most two primes (counting multiplicities), then either G is solvable or G is isomorphic to one of the groups Alt 5 , Sym 5 or an extension of an elementary abelian 2 -group Q of order 1 6 by Alt 5 . In the last case, the set of the co-degrees of the irreducible characters of G is { 1 , 2 2. 3 , 2 2. 5 , 3. 5 , 2 6 }. [ABSTRACT FROM AUTHOR]

Let A = ⊕ α ∈ Γ A α and B = ⊕ α ∈ Γ B α be two commutative rings graded by an arbitrary commutative monoid Γ , let J be a homogeneous ideal of B and f : A → B a graded ring homomorphism. In this paper, we introduce a new graded ring construction called "the graded amalgamation of A with B along J with respect to f " (denoted by A ⋈ f J). Then we investigate, through the properties of the homogeneous components that build the body of this "graded" study, the transfer of the property of graded-coherence to the graded amalgamation A ⋈ f J. [ABSTRACT FROM AUTHOR]

The comaximal graph Γ (R) of a commutative ring R is a simple graph with vertex set R and two distinct vertices u and v of Γ (R) are adjacent if and only if u R + v R = R. In this paper, we find the sharp bounds for the Sombor index for comaximal graphs of integer modulo ring ℤ n and give the corresponding extremal graphs. Also, we find the Sombor eigenvalues and the bounds for the Sombor energy of comaximal graphs of ℤ n . [ABSTRACT FROM AUTHOR]

Given a Hopf algebra H , Brzeziński and Militaru have shown that each braided commutative Yetter–Drinfeld H -module algebra A gives rise to an associative A -bialgebroid structure on the smash product algebra A ♯ H. They also exhibited an antipode map making A ♯ H the total algebra of a Lu's Hopf algebroid over A. However, the published proof that the antipode is an antihomomorphism covers only a special case. In this paper, a complete proof of the antihomomorphism property is exhibited. Moreover, a new generalized version of the construction is provided. Its input is a compatible pair A and A op of braided commutative Yetter–Drinfeld H -module algebras, and output is a symmetric Hopf algebroid A ♯ H ≅ H ♯ A op over  A. This construction does not require that the antipode of H is invertible. [ABSTRACT FROM AUTHOR]

In recent years, SEP elements have been studied by many authors. In this paper, we obtain many new characterizations of SEP elements by using inner inverse, group inverse and Moore–Penrose inverse. Mainly, we construct a lot of equations, study the expression forms of solution to these equations in certain given set. [ABSTRACT FROM AUTHOR]

In this paper, we study generalized right (principally) quasi Baer skew monoid rings. Examples to illustrate and delimit the results are provided. [ABSTRACT FROM AUTHOR]