Search

In this paper, using Gallai's Theorem and the notion of strong resolving graph, we determine the strong metric dimension in annihilating-ideal graph of commutative rings. For reduced rings, an explicit formula is given and for non-reduced rings, under some conditions, strong metric dimension is computed. [ABSTRACT FROM AUTHOR]

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]

This article explores the automorphisms of a Chevalley group of type G2 over a commutative ring R. The authors demonstrate that every automorphism of this group, when R is generated by the invertible elements and the ideal 2R, can be expressed as a combination of ring and inner automorphisms. The paper offers a historical overview of the study of automorphisms of classical groups and Chevalley groups, as well as the methods employed in previous research. The authors introduce definitions and main theorems related to Chevalley groups and their automorphisms. The text focuses on automorphisms of the group Gad(G2, R) and their properties, defining ring automorphisms and inner automorphisms, and establishing standard automorphisms as compositions of these two types. The primary objective is to prove that any automorphism of the group Gad(G2, R) is standard. The text also covers definitions and theorems concerning the localization of rings and modules, isomorphisms of Chevalley groups over fields, and the characteristic subgroup Ead(G2, R) in Gad(G2, R). The main theorem's proof is outlined in several steps, including the embedding of the ring R into a product of its localizations and the mapping of elements under conjugation by an element of Gad(G2, S). Additionally, a lemma is proven that demonstrates the mapping of matrices under conjugation by an element of Gad(G2, S) [Extracted from the article]

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]

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]

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 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 R be a commutative Noetherian ring and I a proper ideal of R. In this paper, we study finitely generated R-modules M with only one non-vanishing local cohomology module H I c (M) where c = c d (I , M) . Let C be a semidualizing R-module. We investigate the conditions under which H I c (M) belongs to either the Auslander class A C (R) or the Bass class B C (R) . [ABSTRACT FROM AUTHOR]

The essential annihilating-ideal graph E G (R) of a commutative unital ring R is a simple graph, whose vertices are non-zero ideals of R with non-zero annihilator and there exists an edge between two distinct vertices I, J if and only if Ann(IJ) has a non-zero intersection with any non-zero ideal of R. In this paper, we show that E G (R) is weakly perfect, if R is Noetherian and an explicit formula for the clique number of E G (R) is given. Moreover, the structures of all rings whose essential annihilating-ideal graphs have chromatic number 2 are fully determined. Among other results, twin-free clique number and edge chromatic number of E G (R) are examined. [ABSTRACT FROM AUTHOR]

We give a new technique for constructing self-dual codes based on a block matrix whose blocks arise from group rings and orthogonal matrices. The technique can be used to construct self-dual codes over finite commutative Frobenius rings of characteristic 2. We give and prove the necessary conditions needed for the technique to produce self-dual codes. We also establish the connection between self-dual codes generated by the new technique and units in group rings. Using the construction together with the building-up construction, we obtain new extremal binary self-dual codes of lengths 64, 66 and 68 and new best known binary self-dual codes of length 80. [ABSTRACT FROM AUTHOR]

In this article, we construct linear codes over the commutative non-unital ring I of size four. We obtain their Lee-weight distributions and study their binary Gray images. Under certain mild conditions, these classes of binary codes are minimal and self-orthogonal. All codes in this article are few-weight codes. Besides, an infinite class of these binary codes consists of distance optimal codes with respect to the Griesmer bound. [ABSTRACT FROM AUTHOR]