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Authors:Changli Ma, Shuxia Liu, Yanan Feng, Yang Zhang, Liwei Zeng Pages: 361 - 374 Abstract: Algebra Colloquium, Volume 29, Issue 03, Page 361-374, September 2022. In this paper, we construct a class of association schemes by using pairs of subspaces of vector spaces and determine their full automorphism groups. Citation: Algebra Colloquium PubDate: 2022-07-26T07:00:00Z DOI: 10.1142/S100538672200027X Issue No:Vol. 29, No. 03 (2022)

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Authors:Yongcai Ren Pages: 375 - 384 Abstract: Algebra Colloquium, Volume 29, Issue 03, Page 375-384, September 2022. An element [math] of a finite group [math] is said to be primary if the order of [math] is a prime power. We define [math] as follows: if [math] is a prime power for every primary element [math] of [math], where [math] is the conjugacy class of [math] in [math], then [math]; if there exists a primary element [math] in [math] such that [math] is divisible by at least two distinct primes, then [math]. In this paper we discuss the influence of the number [math] on the structure of [math]. Citation: Algebra Colloquium PubDate: 2022-07-26T07:00:00Z DOI: 10.1142/S1005386722000281 Issue No:Vol. 29, No. 03 (2022)

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Authors:Honglin Zou, Dragana Cvetković-Ilić, Jianlong Chen, Kezheng Zuo Pages: 385 - 404 Abstract: Algebra Colloquium, Volume 29, Issue 03, Page 385-404, September 2022. In this paper we obtain some equivalent conditions for the core invertibility and EP-ness of [math], [math], [math] and [math], where [math], [math] are projections in different settings, such as ∗-rings, ∗-reducing rings and [math]-algebras. Moreover, several representations for the core inverses of product, difference and sum of two generalized projections are derived. In particular, a number of examples are given to illustrate our results. Citation: Algebra Colloquium PubDate: 2022-07-26T07:00:00Z DOI: 10.1142/S1005386722000293 Issue No:Vol. 29, No. 03 (2022)

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Authors:Ali Majidinya Pages: 405 - 418 Abstract: Algebra Colloquium, Volume 29, Issue 03, Page 405-418, September 2022. For a ring [math] and a strictly totally ordered monoid [math], let [math] be a monoid homomorphism and [math] an [math]-weakly rigid right [math]-module (i.e., for any elements [math], [math] and [math], [math] if and only if [math]), where [math] is the ring of ring endomorphisms of [math]. It is shown that the skew generalized power series module [math] is a principally quasi-Baer module if and only if the annihilator of every submodule generated by an [math]-indexed subset of [math] is generated by an idempotent as a right ideal of [math]. As a consequence we deduce that for an [math]-weakly rigid ring [math], the skew generalized power series ring [math] is right principally quasi-Baer if and only if [math] is right principally quasi-Baer and any [math]-indexed subset of right semicentral idempotents in [math] has a generalized [math]-indexed join in [math]. The range of previous results in this area is expanded by these results. Citation: Algebra Colloquium PubDate: 2022-07-26T07:00:00Z DOI: 10.1142/S100538672200030X Issue No:Vol. 29, No. 03 (2022)

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Authors:Érica Z. Fornaroli Pages: 419 - 426 Abstract: Algebra Colloquium, Volume 29, Issue 03, Page 419-426, September 2022. Let [math] be a finite partially ordered set, [math] an associative unital ring and [math] an endomorphism of [math]. We describe some properties of the skew incidence ring [math] such as invertible elements, idempotents, the Jacobson radical and the center. Moreover, if two skew incidence rings [math] and [math]are isomorphic and the only idempotents of [math] and [math] are the trivial ones, we show that the partially ordered sets [math] and [math] are isomorphic. Citation: Algebra Colloquium PubDate: 2022-07-26T07:00:00Z DOI: 10.1142/S1005386722000311 Issue No:Vol. 29, No. 03 (2022)

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Authors:David Dolžan, Polona Oblak Pages: 427 - 436 Abstract: Algebra Colloquium, Volume 29, Issue 03, Page 427-436, September 2022. We prove that the Laplacian matrix of the total graph of a finite commutative ring with identity has integer eigenvalues and present a recursive formula for computing its eigenvalues and eigenvectors. We also prove that the total graph of a finite commutative local ring with identity is super integral and give an example showing that this is not true for arbitrary rings. Citation: Algebra Colloquium PubDate: 2022-07-26T07:00:00Z DOI: 10.1142/S1005386722000323 Issue No:Vol. 29, No. 03 (2022)

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Authors:Ruifang Chen, Xianhe Zhao Pages: 437 - 442 Abstract: Algebra Colloquium, Volume 29, Issue 03, Page 437-442, September 2022. Let [math] be a finite group and [math] a normal subgroup of [math]. Denote by [math] the graph whose vertices are all distinct [math]-conjugacy class sizes of non-central elements in [math], and two vertices of [math] are adjacent if and only if they are not coprime numbers. We prove that if the center [math] and [math]is [math]-regular for [math], then either a section of [math]is a quasi-Frobenius group or [math] is a complete graph with [math] vertices. Citation: Algebra Colloquium PubDate: 2022-07-26T07:00:00Z DOI: 10.1142/S1005386722000335 Issue No:Vol. 29, No. 03 (2022)

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Authors:A.V. Jayanthan, Rajib Sarkar Pages: 443 - 452 Abstract: Algebra Colloquium, Volume 29, Issue 03, Page 443-452, September 2022. In this article, we obtain an upper bound for the regularity of the binomial edge ideal of a graph whose every block is either a cycle or a clique. As a consequence, we obtain an upper bound for the regularity of binomial edge ideal of a cactus graph. We also identify a certain subclass attaining the upper bound. Citation: Algebra Colloquium PubDate: 2022-07-26T07:00:00Z DOI: 10.1142/S1005386722000347 Issue No:Vol. 29, No. 03 (2022)

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Authors:Ivan Kaygorodov, Pilar Páez-Guillán, Vasily Voronin Pages: 453 - 474 Abstract: Algebra Colloquium, Volume 29, Issue 03, Page 453-474, September 2022. We give a classification of 5- and 6-dimensional complex one-generated nilpotent bicommutative algebras. Citation: Algebra Colloquium PubDate: 2022-07-26T07:00:00Z DOI: 10.1142/S1005386722000359 Issue No:Vol. 29, No. 03 (2022)

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Authors:Dejun Wu, Hui Zhou Pages: 475 - 490 Abstract: Algebra Colloquium, Volume 29, Issue 03, Page 475-490, September 2022. Let [math] and [math] be rings and [math] a [math]-bimodule. If [math] is flat and [math] is finitely generated projective (resp., [math] is finitely generated projective and [math] is flat), then the characterizations of level modules and Gorenstein AC-projective modules (resp., absolutely clean modules and Gorenstein AC-injective modules) over the formal triangular matrix ring [math] are given. As applications, it is proved that every Gorenstein AC-projective left [math]-module is projective if and only if each Gorenstein AC-projective left [math]-module and [math]-module is projective, and every Gorenstein AC-injective left [math]-module is injective if and only if each Gorenstein AC-injective left [math]-module and [math]-module is injective. Moreover, Gorenstein AC-projective and AC-injective dimensions over the formal triangular matrix ring [math] are studied. Citation: Algebra Colloquium PubDate: 2022-07-26T07:00:00Z DOI: 10.1142/S1005386722000360 Issue No:Vol. 29, No. 03 (2022)

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Authors:Matthew Ondrus, Emilie Wiesner Pages: 491 - 508 Abstract: Algebra Colloquium, Volume 29, Issue 03, Page 491-508, September 2022. The Lie algebra [math] may be regarded in a natural way as a subalgebra of the infinite-dimensional Virasoro Lie algebra, so it is natural to consider connections between the representation theory of the two algebras. In this paper, we explore the restriction to [math] of certain induced modules for the Virasoro algebra. Specifically, we consider Virasoro modules induced from so-called polynomial subalgebras, and we show that the restriction of these modules results in twisted versions of familiar modules such as Verma modules and Whittaker modules. Citation: Algebra Colloquium PubDate: 2022-07-26T07:00:00Z DOI: 10.1142/S1005386722000372 Issue No:Vol. 29, No. 03 (2022)

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Authors:Carmelo Cisto Pages: 509 - 526 Abstract: Algebra Colloquium, Volume 29, Issue 03, Page 509-526, September 2022. In this paper we introduce a particular semigroup transform [math] that fixes the invariants involved in Wilf's conjecture, except the embedding dimension. It also allows one to arrange the set of non-ordinary and non-irreducible numerical semigroups in a family of rooted trees. In addition, we study another transform, having similar features, that has been introduced by Bras-Amorós, and we make a comparison of them. In particular, we study the behavior of the embedding dimension under the action of such transforms, providing some consequences concerning Wilf's conjecture. Citation: Algebra Colloquium PubDate: 2022-07-26T07:00:00Z DOI: 10.1142/S1005386722000384 Issue No:Vol. 29, No. 03 (2022)

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Authors:Long Chen, Zongbing Lin, Qianrong Tan Pages: 527 - 540 Abstract: Algebra Colloquium, Volume 29, Issue 03, Page 527-540, September 2022. Let [math], [math] and [math] be positive integers with[math], [math] be an integer-valued arithmetic function, and the set [math] of [math] distinct positive integers be a divisor chain such that [math]. We first show that the matrix [math] having [math] evaluated at the [math]th power [math] of the greatest common divisor of [math] and [math] as its [math]-entry divides the GCD matrix [math] in the ring [math] of [math] matrices over integers if and only if [math] and [math] divides [math] for any integer [math] with [math]. Consequently, we show that the matrix [math] having [math] evaluated at the [math]th power [math] of the least common multiple of [math] and [math] as its [math]-entry divides the matrix [math] in the ring [math] if and only if [math] and [math] divides [math] for any integer [math] with[math]. Finally, we prove that the matrix [math] divides the matrix [math] in the ring [math] if and only if [math] and [math] for any integer [math] with [math]. Our results extend and strengthen the theorems of Hong obtained in 2008. Citation: Algebra Colloquium PubDate: 2022-07-26T07:00:00Z DOI: 10.1142/S1005386722000396 Issue No:Vol. 29, No. 03 (2022)