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Let I be the commutative non-unital ring of order 4 defined by generators and relations. I = a , b ∣ 2 a = 2 b = 0 , a 2 = b , a b = 0. Alahmadi et al. have classified QSD codes, Type IV codes (QSD codes with even weights) and quasi-Type IV codes (QSD codes with even torsion code) over I up to lengths n = 6 , and suggested two building-up methods for constructing QSD codes. In this paper, we construct more QSD codes, Type IV codes and quasi-Type IV codes for lengths n = 7 and 8, and describe five new variants of the two building-up construction methods. We find that when n = 8 there is at least one QSD code with minimun distance 4, which attains the highest minimum distance so far, and we give a generator matrix for the code. We also describe some QSD codes, Type IV codes and quasi-Type IV codes with new weight distributions. [ABSTRACT FROM AUTHOR]

We present the following constructions: extension of a trivial quandle by a group with a nontrivial abelian subgroup, a quandle over a noncommutative group by an arbitrary antiautomorphism, a quandle presenting from a generalized Alexander quandle with the replacement of the automorphism by a central antiautomorphism, and a quandle over an -dimensional module depending on parameters with in a commutative unital ring. [ABSTRACT FROM AUTHOR]

Since the beginning of research on graph related to ring, many graphs have been constructed by the concept of rings to investigate the relation between graph theory and ring theory. The unity product graph associated with a ring R is a graph which is obtained by setting all unit elements of R as the vertex set and two distinct elements ri and rj are joined by an edge if and only if ri · rj = e. A subset C of the vertex set in a unity product graph is called a perfect code if for all ri 2 C, the unions of their closed neighbourhoods, S1(ri) yield the whole vertex set in such a way that any of two closed neighbourhoods have no common elements. In this paper, we determine different order perfect codes in unity product graphs associated with some commutative rings with identity. We classify the commutative rings with identity in which their associated unity product graphs admit the trivial (order 1) and non-trivial perfect codes. In addition, it is proved that the complement unity product graphs associated with the commutative rings with identity admit only the trivial perfect code. [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]

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]

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]

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]

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]

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]

An h-local domain is a domain for which each nonzero prime ideal is contained in a unique maximal ideal and each nonzero element has finite character. In his dissertation, A. Omairi generalizes the notion of an h-local domain to rings with zero-divisors by restricting the definition to regular ideals. In this article, we give a characterization of when C(X) the ring of continuous real-valued functions on a space X is an h-local ring. We are left with more questions than answers. [ABSTRACT FROM AUTHOR]

We examine those matrix rings whose entries lie in periodic rings equipped with some additional properties. Specifically, we prove that the famous Diesl's question whether or not a ring R being nil-clean implies that the matrix ring M n (R) over R is also nil-clean for all n ≥ 1 is paralleling to the corresponding implication for (abelian, local) periodic rings. Besides, we study when the endomorphism ring E (G) of an abelian group G is periodic. Concretely, we establish that E (G) is periodic exactly when G is finite as well as we find a complete necessary and sufficient condition when the endomorphism ring over an abelian group is strongly m-nil clean for some natural number m thus refining an "old" result concerning strongly nil-clean endomorphism rings. Responding to a question when a group ring is periodic, we show that if R is a right (resp., left) perfect periodic ring and G is a locally finite group, then the group ring RG is periodic, too. We finally find some criteria under certain conditions when the tensor product of two periodic algebras over a commutative ring is again periodic. In addition, some other sorts of rings very close to periodic rings, namely the so-called weakly periodic rings, are also investigated. [ABSTRACT FROM AUTHOR]

Let m, n be two nonzero fixed positive integers, R a prime ring with the right Martindale quotient ring Q, G and H two generalized skew derivations of R with the same associated automorphism α of R. If G (x m + n) = H (x m) x n is an identity for R, then either R is commutative or there exists q ∈ Q such that G (x) = H (x) = q x , for any x ∈ R . [ABSTRACT FROM AUTHOR]

Let R be a commutative ring and Z (R) be its zero-divisors set. The zero-divisor graph of R , denoted by Γ (R) , is an undirected graph with vertex set Z (R) ∗ = Z (R) ∖ { 0 } and two distinct vertices a and b are adjacent if and only if a b = 0. In this paper, for n = p R q S where p and q are primes (p < q) and R and S are positive integers, we calculate the A α spectrum of the graphs Γ (ℤ n). [ABSTRACT FROM AUTHOR]

Let R be a semiprime ring and α be an automorphism of R. A mapping F : R → R (not necessarily additive) is called multiplicative generalized (α, α)- derivation if there exists a unique (α, α)-derivation d of R such that F(xy) = F(x)α(y) + α(x)d(y) for all x, y = R. In the present paper, we intend to study several algebraic identities involving multiplicative generalized (α, α)-derivations on appropriate subsets of semiprime rings and collect the information about the commutative structure of these rings. [ABSTRACT FROM AUTHOR]

This study investigates the application of graph theory in analyzing the zero divisor graph of a commutative ring, with a specific focus on its connection to the topological index. For an undirected graph G with consists of a non-empty set of vertices, V, and a set of edges, E, the first general Zagreb index is defined as a graph invariant that measures the sum of the degree of each vertex to the power of α ≠ 0. Meanwhile, the zero divisor graph G of the commutative ring, R is the (undirected) graph with vertices the zero-divisors of R, and distinct vertices a and b are adjacent if and only if ab = 0. In this paper, the general formulas of the first general Zagreb index of the zero divisor graph for the ring of integers modulo pqk are computed for the cases δ = 1, 2, and 3. This research focuses on the ring defined as the integers modulo pqk, where k is a positive integer, p and q are primes p < q. Two examples are given to demonstrate the main findings. [ABSTRACT FROM AUTHOR]

Let Γ be a group, A be a Γ-graded commutative ring with unity 1, and D a graded A-module. In this paper, we introduce the concept of graded weakly Jgr-semiprime submodules as a generalization of graded weakly semiprime submodules. We study several results concerning of graded weakly Jgr-semiprime submodules. For example, we give a characterization of graded weakly Jgr-semiprime submodules. Also, we find some relations between graded weakly Jgr-semiprime submodules and graded weakly semiprime submodules. In addition, the necessary and sufficient condition for graded submodules to be graded weakly Jgr-semiprime submodules are investigated. A proper graded submodule U of D is said to be a graded weakly Jgr-semiprime submodule of D if whenever rg ∈ h(A); mh ∈ h(D) and n ∈ Z+ with 0 ≠ rngmh ∈ U, then rgmh ∈ U + Jgr(D), where Jgr(D) is the graded Jacobson radical of D. [ABSTRACT FROM AUTHOR]

This paper studies λ-constacyclic codes and skew λ-constacyclic codes over the finite commutative non-chain ring R = Z4 + uZ4 + u²Z4 with u³ = 0 for λ = (1 + 2u + 2u²) and (3 + 2u + 2u²). We introduce distinct Gray maps and show that the Gray images of λ-constacyclic codes are cyclic, quasi-cyclic, and permutation equivalent to quasi-cyclic codes over Z4. It is also shown that the Gray images of skew λ-constacyclic codes are quasi-cyclic codes of length 2n and index 2 over Z4. Moreover, the structure of λ-constacyclic codes of odd length n over the ring R is determined and give some suitable examples. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity and let M be an R-module. The purpose of this paper is to introduce and investigate the submodules of an R-module M which satisfy the dual of Property A, the dual of strong Property A, and the dual of proper strong Property A. Moreover, a submodule N of M which satisfy Property SJ(N) and Property IMJ(N) will be introduced and investigated. [ABSTRACT FROM AUTHOR]

Let G be a group with identity e. Let R be a G-graded commutative ring. In this paper, we introduce and study the concept of graded uniformly n-ideal (in short, gr-u-n-ideal). Many results concerning gr-u-n-ideals are obtained. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity. For positive integers m and n, Anderson and Badawi (Journal of Algebra and Its Applications 16(1):1750013 (21 pp), 2017) defined an ideal I of a ring R to be an (m,n)-closed if whenever x m ∈ I , then x n ∈ I . In this paper we define and study a new generalization of the class of (m,n)-closed ideals which is the class of quasi (m,n)-closed ideals. A proper ideal I is called quasi (m,n)-closed in R if for x ∈ R , x m ∈ I implies either x n ∈ I or x m - n ∈ I . That is, I is quasi (m,n)-closed in R if and only if I is either (m, n)-closed or ( m , m - n )-closed in R. We justify several properties and characterizations of quasi (m,n)-closed ideals with many supporting examples. Furthermore, we investigate quasi (m,n)-closed ideals under various contexts of constructions such as direct products, localizations and homomorphic images. Finally, we discuss the behavior of this class of ideals in idealization rings. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity, M a multiplication R-module, and T(M)* the set of non-zero torsion elements of M. We consider two graphs, the torsion graph and the annihilator graph of M that have T(M)* as their set of vertices, and investigate the cases when these graphs are stars. The graph theoretic properties are reflected in the ring theoretic properties and vice versa. If a ring is considered as a module on itself, then the module is a multiplication module. Hence, our results directly generalize results about rings. [ABSTRACT FROM AUTHOR]

In this paper, we study the action of diamond operators on Hilbert modular forms with coefficients in a general commutative ring. In particular, we generalize a result of Chai on the surjectivity of the constant term map for Hilbert modular forms with nebentype to the setting of group ring valued modular forms. As an application, we construct certain Hilbert modular forms required for Dasgupta–Kakde's proof of the Brumer–Stark conjecture at odd primes. Since the forms required for the Brumer–Stark conjecture live on the non-PEL Shimura variety associated to the reductive group G = Res F / Q (GL 2) , as opposed to the PEL Shimura variety associated to the subgroup G ∗ ⊂ G studied by Chai, we give a detailed explanation of theory of algebraic diamond operators for G, as well as how the theory of toroidal and minimal compactifications for G may be deduced from the analogous theory for G ∗ . [ABSTRACT FROM AUTHOR]

Spectral curves are algebraic curves associated to commutative subalgebras of rings of ordinary differential operators (ODOs). Their origin is linked to the Korteweg–de Vries equation and to seminal works on commuting ODOs by I. Schur and Burchnall and Chaundy. They allow the solvability of the spectral problem L y = λ y , for an algebraic parameter λ and an algebro-geometric ODO L, whose centralizer is known to be the affine ring of an abstract spectral curve Γ. In this work, we use differential resultants to effectively compute the defining ideal of the spectral curve Γ , defined by the centralizer of a third-order differential operator L, with coefficients in an arbitrary differential field of zero characteristic. For this purpose, defining ideals of planar spectral curves associated to commuting pairs are described as radicals of differential elimination ideals. In general, Γ is a non-planar space curve and we provide the first explicit example. As a consequence, the computation of a first-order right factor of L − λ becomes explicit over a new coefficient field containing Γ. Our results establish a new framework appropriate to develop a Picard–Vessiot theory for spectral problems. [ABSTRACT FROM AUTHOR]

Over algebraically closed fields of arbitrary characteristic, we prove a general multiplicity-freeness theorem for finitely-generated modules with commutative endomorphism rings. Using more lenient versions of projectivity and injectivity for modules, this gives a generalization of Gelfand's criterion on the commutativity of Hecke algebras for Gelfand pairs and multiplicity-free triples. For representations of finite and profinite groups, Gelfand pairs over the complex numbers are therefore also Gelfand pairs over the algebraic closure of any finite field. Applications include the uniqueness of Whittaker models of Gelfand–Graev representations in arbitrary characteristic and the uniqueness of modular trilinear forms on irreducible representations of quaternion division algebras over local fields. [ABSTRACT FROM AUTHOR]

We characterize ultragraph Leavitt path algebras that are Rickart, locally Rickart, graded Rickart, and graded Rickart *-rings. We also characterize ultragraph Leavitt path algebras that are Baer, locally Baer, graded Baer, Baer *-rings, and combinations of these. These characterizations build on and generalize the work of Hazrat and Vaš on Leavitt path algebras over fields to ultragraph Leavitt path algebras over semi-simple commutative unital rings. [ABSTRACT FROM AUTHOR]

Let G be a group. Let R be a G-graded commutative ring and let M be a graded R-module. A proper graded submodule Q of M is called a graded quasi-primary submodule if whenever r ∈ h ⁢ (R) and m ∈ h ⁢ (M) with r ⁢ m ∈ Q , then either r ∈ Gr ((Q : R M)) or m ∈ Gr M ⁡ (Q) . The graded quasi-primary spectrum qp. Spec g (M) is defined to be the set of all graded quasi-primary submodules of M. In this paper, we introduce and study a topology on qp. Spec g (M) , called the quasi-Zariski topology, and investigate the properties of this topology and some conditions under which ( qp. Spec g (M) , q. τ g) is a Noetherian, spectral space. [ABSTRACT FROM AUTHOR]

We study the derivations of the incidence algebra I (X , R) , where X is a preordered set and R is an algebra over some commutative ring T. A satisfactory description of the T-module of derivations and the T-module of outer derivations of this algebra is given. [ABSTRACT FROM AUTHOR]

We observe that Beck modules for a commutative monoid are exactly modules over a graded commutative ring associated to the monoid. Under this identification, the Quillen cohomology of commutative monoids is a special case of the André–Quillen cohomology for graded commutative rings, generalizing a result of Kurdiani and Pirashvili. To verify this we develop the necessary grading formalism. The partial cochain complex developed by Pierre Grillet for computing Quillen cohomology appears as the start of a modification of the Harrison cochain complex suggested by Michael Barr. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with 1 and M be a left unitary R-module. In this paper, we give a generalization for the notions of compressible (retractable) module. As well as, we study closed (St-closed) compressible and closed (St-closed) retractable. Furthermore, some of their advantages, properties, categorizations and instances have been given. Finally, we study the relation between them. [ABSTRACT FROM AUTHOR]

We construct examples of bounded below, noncontractible, acyclic complexes of finitely generated projective modules over some rings S, as well as bounded above, noncontractible, acyclic complexes of injective modules. The rings S are certain rings of infinite matrices with entries in the rings of commutative polynomials or formal power series in infinitely many variables. In the world of comodules or contramodules over coalgebras over fields, similar examples exist over the cocommutative symmetric coalgebra of an infinite-dimensional vector space. A simpler, universal example of a bounded below, noncontractible, acyclic complex of free modules with one generator, communicated to the author by Canonaco, is included at the end of the paper. [ABSTRACT FROM AUTHOR]

In this paper, we continue the study of Hurwitz series over a commutative unital ring that was begun in [W. Keigher, On the ring of Hurwitz series, Commun. Algebra 25 (1997) 1845–1859; W. Keigher and F. Pritchard, Hurwitz series as formal functions, J. Pure Appl. Algebra 146 (2000) 291–304]. In particular, we introduce the notion of multivariate Hurwitz series. The underlying idea is that multivariate Hurwitz series are to Hurwitz series as studied in [W. Keigher, On the ring of Hurwitz series, Commun. Algebra 25 (1997) 1845–1859; W. Keigher and F. Pritchard, Hurwitz series as formal functions, J. Pure Appl. Algebra 146 (2000) 291–304] as formal power series in several indeterminates are to formal power series in only one indeterminate. The elementary aspects of the theory follow along the lines of [W. Keigher, On the ring of Hurwitz series, Commun. Algebra 25 (1997) 1845–1859; W. Keigher and F. Pritchard, Hurwitz series as formal functions, J. Pure Appl. Algebra 146 (2000) 291–304]. The treatment of substitution and divided powers introduces special problems not encountered in [W. Keigher and F. Pritchard, Hurwitz series as formal functions, J. Pure Appl. Algebra 146 (2000) 291–304] and requires special attention to subtle details. However, we are able to establish analogous results. With substitution and divided powers in place, we construct and study the analog to the so called inner transformations of [S. Bochner and W. T. Martin, Several Complex Variables (Princeton University Press, 1948)]. Finally, we are able to establish analogs to many of the fundamental results of single and multivariate calculus. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with nonzero identity. A proper ideal I of R is called a 1-absorbing prime ideal (respectively, 1-absorbing primary ideal) if whenever nonunit elements a , b , c ∈ R with a b c ∈ I , then a b ∈ I or c ∈ I (respectively, a b ∈ I or c ∈ I ). The purpose of this paper is to study the transfer of certain 1-absorbing-like properties to amalgamation of A with B along J with respect to f (denoted by A ⋈ f J), introduced and studied by D'Anna, Finocchiaro and Fontana. Our results provide new techniques for the construction of new original examples satisfying the above-mentioned properties. [ABSTRACT FROM AUTHOR]

For a commutative local ring R, we know that every ideal is a direct sum of local modules if and only if the unique maximal ideal M of R is of the form M = R x ⊕ R y ⊕ (⊕ γ ∈ Γ R s γ) , where R / Ann (x) and R / Ann (y) are principal ideal rings and s γ 2 = 0 for all γ's. This motivates us to determine a commutative ring, not necessarily local, in which every (proper) ideal is a direct sum of local modules. We prove that such a ring is exactly a finite direct product of local rings R with the unique maximal ideal M in the above mentioned form. Moreover, we characterize more precisely certain commutative rings such as Noetherian rings, reduced rings, Rickart rings, semi-Artinian rings and self-injective rings, in which every (proper) ideal is a direct sum of local modules. [ABSTRACT FROM AUTHOR]

Let R 1 ⊆ R 2 be commutative rings with unity and let S 1 ⊆ S 2 be torsion-free abelian monoids. We give a characterization of when the extension R 1 [ S 1 ] ⊆ R 2 [ S 2 ] of monoid rings is a root extension. For torsion-free, ≤ -cancellative, strictly ordered abelian monoids (S 1 , ≤) ⊆ (S 2 , ≤) , we also give a characterization of when the extension [ ​ [ R 1 S 1 , ≤ ] ​ ] ⊆ [ ​ [ R 2 S 2 , ≤ ] ​ ] of generalized power series rings is a root extension. [ABSTRACT FROM AUTHOR]

For an edge ideal I(G) of a simple graph G , we study the N -graded Betti numbers that appear in the linear strand of the minimal free resolution of I (Γ (Z n)) , where Γ (Z n) is the comaximal graph of the integral modulo ring Z n . We show that the extremal Betti number of I (Γ (Z n)) is ϕ (n) , where ϕ (n) is the Euler's totient function and thereby we obtain a large class of edge ideals with even extremal Betti numbers. We find the regularity (Castelnuovo-Mumford) and the projective dimension of these ideals. Moreover, we exhibit explicit formulae that determine all the N -graded Betti numbers in the linear strand of the minimal free resolution of I (Γ (Z n)) for certain values of n. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with unity and M be an R-module. The aim of this article is to introduce and investigate certain properties of a new class of submodules namely e-secondary submodules. A submodule N of M is said to be e-secondary if the endomorphism given by multiplication by a, a ∈ R, that is, f : N → N such that f(N) = aN, then either aN ≤e M or atN = 0, for some t ∈ N . This notion is then further extended to introduce fully e-secondary submodules of a module. [ABSTRACT FROM AUTHOR]

Nicholson introduced clean rings, where each element of ring is the sum of an idempotent and unit element. Motivated by this structure we introduce here two classes of rings, qclean and almost qclean ring. In this article we focus to study the fundamental properties of both qclean and almost qclean rings, also proved that if a ring R is (almost) qclean, then power series ring R[[x]] is (almost) qclean. Apart from this we established for a commutative torsionless cancellative monoid T, a commutative graded ring R = ⊕β∈T Tβ is almost qclean, if T0 is almost qclean and each Tβ is torsion-free T0 module. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring and let N be a non-zero finitely generated R-module. In this note, the main result asserts that for any N-proper ideal a of R, the a-symbolic topology on N is linearly equivalent to the a-adic topology on N if and only if, for every p Supp(N), AssRp Np consists of a single prime ideal and dimN = 1. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of ring called regular S-Noetherian ring, which is a weak version of S-Noetherian ring property. Any S-Noetherian ring is naturally a regular S-Noetherian ring, and in the domain context, these two forms coincide. We study the transfer of this notion to various context of commutative ring extensions such as direct product, trivial ring extensions and amalgamated duplication of a ring along an ideal. Our results generate new families of examples of non-S-Noetherian regular S-Noetherian rings. [ABSTRACT FROM AUTHOR]

Let A be a commutative ring. In this paper, we introduce and study the concept of 2J-submodule A submodule U of a A-module Tis called a 2J-submodule) if for all a ∈ A and t ∈ T whenever at ∈ U with a² ∉ (j(A)T:T), then t ∈ U In this work examine the characteristics of the 2J-submodule as a generalization of the J-submodule. This paper provides various characterizations and properties of 2j-submodule. [ABSTRACT FROM AUTHOR]

In this paper, we derive a mass formula for the self-orthogonal codes and self-dual codes over a non-commutative non-unitary ring, namely, E p = a , b | p a = p b = 0 , a 2 = a , b 2 = b , a b = a , b a = b , where a ≠ b and p is any odd prime. We also give a classification of self-orthogonal codes and self-dual codes over E p , where p = 3 , 5 , and 7, in short lengths. [ABSTRACT FROM AUTHOR]

The build-up method is a powerful class of propagation rules that generate self-dual codes over finite fields and unitary rings. Recently, it was extended to non-unitary rings of order 4, to generate quasi self-dual codes. In the present paper, we introduce three such propagation rules to generate self-orthogonal, self-dual and quasi self-dual codes over a special non-unitary ring of order 9. As an application, we classify the three categories of codes completely in length at most 3, and partially in lengths 4 and 5, up to monomial equivalence. [ABSTRACT FROM AUTHOR]

We prove a first part of the standard description of groups H lying between an exterior power of an elementary group $ {m}\operatorname {E}_n(R) $ m E n ⁡ (R) and a general linear group $ \operatorname {GL}_{\binom {n}{m}}(R) $ GL (n m) ⁡ (R) for a commutative ring $ R, 2 \in R^{*} $ R , 2 ∈ R ∗ and $ n \geqslant 3m $ n ⩾ 3 m. The description uses the classical notion of a level: for every group H we find a unique ideal A of the ground ring R, which describes H. [ABSTRACT FROM AUTHOR]

In this paper, we extend our investigation about the generalized comaximal graph introduced in Biswas et al. (Discrete Math Algorithms Appl 11(1):1950013, 2019a). The generalized comaximal graph is defined as follows: given a finite commutative ring R, the generalized comaximal graph G(R) is an undirected graph with its vertex set comprising elements of R and two distinct vertices u, v are adjacent if and only if there exists a non-zero idempotent e ∈ R such that u R + v R = e R . In this study, we focus on identifying the rings R for which the graph G(R) exhibits planarity. Moreover, we provide a characterization of the class of ring for which G(R) is toroidal, denoted by γ (G (R)) = 1 . Furthermore, we also evaluate the energy of the graph G(R). Finally, we demonstrate that the graph G(R) is always Hamiltonian for any finite commutative ring R. [ABSTRACT FROM AUTHOR]

Let $G= N\rtimes H$ be a locally compact group which is a semi-direct product of a closed normal subgroup N and a closed subgroup H. The Bohr compactification ${\rm Bohr}(G)$ and the profinite completion ${\rm Prof}(G)$ of G are, respectively, isomorphic to semi-direct products $Q_1 \rtimes {\rm Bohr}(H)$ and $Q_2 \rtimes {\rm Prof}(H)$ for appropriate quotients $Q_1$ of ${\rm Bohr}(N)$ and $Q_2$ of ${\rm Prof}(N).$ We give a precise description of $Q_1$ and $Q_2$ in terms of the action of H on appropriate subsets of the dual space of N. In the case where N is abelian, we have ${\rm Bohr}(G)\cong A \rtimes {\rm Bohr}(H)$ and ${\rm Prof}(G)\cong B \rtimes {\rm Prof}(H),$ where A (respectively B) is the dual group of the group of unitary characters of N with finite H -orbits (respectively with finite image). Necessary and sufficient conditions are deduced for G to be maximally almost periodic or residually finite. We apply the results to the case where $G= \Lambda\wr H$ is a wreath product of discrete groups; we show in particular that, in case H is infinite, ${\rm Bohr}(\Lambda\wr H)$ is isomorphic to ${\rm Bohr}(\Lambda^{\rm Ab}\wr H)$ and ${\rm Prof}(\Lambda\wr H)$ is isomorphic to ${\rm Prof}(\Lambda^{\rm Ab} \wr H),$ where $\Lambda^{\rm Ab}=\Lambda/ [\Lambda, \Lambda]$ is the abelianisation of $\Lambda.$ As examples, we compute ${\rm Bohr}(G)$ and ${\rm Prof}(G)$ when G is a lamplighter group and when G is the Heisenberg group over a unital commutative ring. [ABSTRACT FROM AUTHOR]

We generalize Grothendieck's generic freeness lemma to modules over semi-constructible extensions of normally bounded domains, extending previous results of Irving, McConnell, Robson, Artin, Small and Zhang, in which the base ring is commutative. [ABSTRACT FROM AUTHOR]

An order is a commutative ring that as an abelian group is finitely generated and free. A commutative ring is reduced if it has no non-zero nilpotent elements. In this paper we use a new tool, namely, the fact that every reduced order has a universal grading, to answer questions about realizing orders as group rings. In particular, we address the Isomorphism Problem for group rings in the case where the ring is a reduced order. We prove that any non-zero reduced order R can be written as a group ring in a unique "maximal" way, up to isomorphism. More precisely, there exist a ring A and a finite abelian group G , both uniquely determined up to isomorphism, such that R ≅ A [ G ] as rings, and such that if B is a ring and H is a group, then R ≅ B [ H ] as rings if and only if there is a finite abelian group J such that B ≅ A [ J ] as rings and J × H ≅ G as groups. Computing A and G for given R can be done by means of an algorithm that is not quite polynomial-time. We also give a description of the automorphism group of R in terms of A and G. [ABSTRACT FROM AUTHOR]