Pub Date : 2022-07-26DOI: 10.1142/s1005386722000281
Yongcai Ren
An element [Formula: see text] of a finite group [Formula: see text] is said to be primary if the order of [Formula: see text] is a prime power. We define [Formula: see text] as follows: if [Formula: see text] is a prime power for every primary element [Formula: see text] of [Formula: see text], where [Formula: see text] is the conjugacy class of [Formula: see text] in [Formula: see text], then [Formula: see text]; if there exists a primary element [Formula: see text] in [Formula: see text] such that [Formula: see text] is divisible by at least two distinct primes, then [Formula: see text]. In this paper we discuss the influence of the number [Formula: see text] on the structure of [Formula: see text].
{"title":"An Arithmetical Condition on the Sizes of Conjugacy Classes of a Finite Group","authors":"Yongcai Ren","doi":"10.1142/s1005386722000281","DOIUrl":"https://doi.org/10.1142/s1005386722000281","url":null,"abstract":"An element [Formula: see text] of a finite group [Formula: see text] is said to be primary if the order of [Formula: see text] is a prime power. We define [Formula: see text] as follows: if [Formula: see text] is a prime power for every primary element [Formula: see text] of [Formula: see text], where [Formula: see text] is the conjugacy class of [Formula: see text] in [Formula: see text], then [Formula: see text]; if there exists a primary element [Formula: see text] in [Formula: see text] such that [Formula: see text] is divisible by at least two distinct primes, then [Formula: see text]. In this paper we discuss the influence of the number [Formula: see text] on the structure of [Formula: see text].","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73990366","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-26DOI: 10.1142/s1005386722000323
David Dolžan, Polona Oblak
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.
{"title":"Total Graphs Are Laplacian Integral","authors":"David Dolžan, Polona Oblak","doi":"10.1142/s1005386722000323","DOIUrl":"https://doi.org/10.1142/s1005386722000323","url":null,"abstract":"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.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79610409","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-26DOI: 10.1142/s100538672200027x
Changli Ma, Shuxia Liu, Yanan Feng, Yang Zhang, Liwei Zeng
In this paper, we construct a class of association schemes by using pairs of subspaces of vector spaces and determine their full automorphism groups.
本文利用向量空间的子空间对构造了一类关联方案，并确定了它们的满自同构群。
{"title":"A Class of Association Schemes and Their Automorphisms","authors":"Changli Ma, Shuxia Liu, Yanan Feng, Yang Zhang, Liwei Zeng","doi":"10.1142/s100538672200027x","DOIUrl":"https://doi.org/10.1142/s100538672200027x","url":null,"abstract":"In this paper, we construct a class of association schemes by using pairs of subspaces of vector spaces and determine their full automorphism groups.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91131656","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-26DOI: 10.1142/s1005386722000293
Honglin Zou, D. Cvetković-Ilić, Jianlong Chen, Kezheng Zuo
In this paper we obtain some equivalent conditions for the core invertibility and EP-ness of [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text], [Formula: see text] are projections in different settings, such as ∗-rings, ∗-reducing rings and [Formula: see text]-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.
{"title":"Characterizations for the Core Invertibility and EP-ness Involving Projections","authors":"Honglin Zou, D. Cvetković-Ilić, Jianlong Chen, Kezheng Zuo","doi":"10.1142/s1005386722000293","DOIUrl":"https://doi.org/10.1142/s1005386722000293","url":null,"abstract":"In this paper we obtain some equivalent conditions for the core invertibility and EP-ness of [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text], [Formula: see text] are projections in different settings, such as ∗-rings, ∗-reducing rings and [Formula: see text]-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.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88200766","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-26DOI: 10.1142/s1005386722000372
Matthew Ondrus, Emilie Wiesner
The Lie algebra [Formula: see text] 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 [Formula: see text] 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.
{"title":"The Restriction of Polynomial Modules for the Virasoro Algebra to sl2(C)","authors":"Matthew Ondrus, Emilie Wiesner","doi":"10.1142/s1005386722000372","DOIUrl":"https://doi.org/10.1142/s1005386722000372","url":null,"abstract":"The Lie algebra [Formula: see text] 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 [Formula: see text] 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.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77546320","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-26DOI: 10.1142/s100538672200030x
A. Majidinya
For a ring [Formula: see text] and a strictly totally ordered monoid [Formula: see text], let [Formula: see text] be a monoid homomorphism and [Formula: see text] an [Formula: see text]-weakly rigid right [Formula: see text]-module (i.e., for any elements [Formula: see text], [Formula: see text] and [Formula: see text], [Formula: see text] if and only if [Formula: see text]), where [Formula: see text] is the ring of ring endomorphisms of [Formula: see text]. It is shown that the skew generalized power series module [Formula: see text] is a principally quasi-Baer module if and only if the annihilator of every submodule generated by an [Formula: see text]-indexed subset of [Formula: see text] is generated by an idempotent as a right ideal of [Formula: see text]. As a consequence we deduce that for an [Formula: see text]-weakly rigid ring [Formula: see text], the skew generalized power series ring [Formula: see text] is right principally quasi-Baer if and only if [Formula: see text] is right principally quasi-Baer and any [Formula: see text]-indexed subset of right semicentral idempotents in [Formula: see text] has a generalized [Formula: see text]-indexed join in [Formula: see text]. The range of previous results in this area is expanded by these results.
{"title":"On the P.Q.-Baer Skew Generalized Power Series Modules","authors":"A. Majidinya","doi":"10.1142/s100538672200030x","DOIUrl":"https://doi.org/10.1142/s100538672200030x","url":null,"abstract":"For a ring [Formula: see text] and a strictly totally ordered monoid [Formula: see text], let [Formula: see text] be a monoid homomorphism and [Formula: see text] an [Formula: see text]-weakly rigid right [Formula: see text]-module (i.e., for any elements [Formula: see text], [Formula: see text] and [Formula: see text], [Formula: see text] if and only if [Formula: see text]), where [Formula: see text] is the ring of ring endomorphisms of [Formula: see text]. It is shown that the skew generalized power series module [Formula: see text] is a principally quasi-Baer module if and only if the annihilator of every submodule generated by an [Formula: see text]-indexed subset of [Formula: see text] is generated by an idempotent as a right ideal of [Formula: see text]. As a consequence we deduce that for an [Formula: see text]-weakly rigid ring [Formula: see text], the skew generalized power series ring [Formula: see text] is right principally quasi-Baer if and only if [Formula: see text] is right principally quasi-Baer and any [Formula: see text]-indexed subset of right semicentral idempotents in [Formula: see text] has a generalized [Formula: see text]-indexed join in [Formula: see text]. The range of previous results in this area is expanded by these results.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81618476","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-26DOI: 10.1142/s1005386722000396
Long Chen, Zongbing Lin, Qianrong Tan
Let [Formula: see text], [Formula: see text] and [Formula: see text] be positive integers with[Formula: see text], [Formula: see text] be an integer-valued arithmetic function, and the set [Formula: see text] of [Formula: see text] distinct positive integers be a divisor chain such that [Formula: see text]. We first show that the matrix [Formula: see text] having [Formula: see text] evaluated at the [Formula: see text]th power [Formula: see text] of the greatest common divisor of [Formula: see text] and [Formula: see text] as its [Formula: see text]-entry divides the GCD matrix [Formula: see text] in the ring [Formula: see text] of [Formula: see text] matrices over integers if and only if [Formula: see text] and [Formula: see text] divides [Formula: see text] for any integer [Formula: see text] with [Formula: see text]. Consequently, we show that the matrix [Formula: see text] having [Formula: see text] evaluated at the [Formula: see text]th power [Formula: see text] of the least common multiple of [Formula: see text] and [Formula: see text] as its [Formula: see text]-entry divides the matrix [Formula: see text] in the ring [Formula: see text] if and only if [Formula: see text] and [Formula: see text] divides [Formula: see text] for any integer [Formula: see text] with[Formula: see text]. Finally, we prove that the matrix [Formula: see text] divides the matrix [Formula: see text] in the ring [Formula: see text] if and only if [Formula: see text] and [Formula: see text] for any integer [Formula: see text] with [Formula: see text]. Our results extend and strengthen the theorems of Hong obtained in 2008.
{"title":"Divisibility Properties of Power Matrices Associated with Arithmetic Functions on a Divisor Chain","authors":"Long Chen, Zongbing Lin, Qianrong Tan","doi":"10.1142/s1005386722000396","DOIUrl":"https://doi.org/10.1142/s1005386722000396","url":null,"abstract":"Let [Formula: see text], [Formula: see text] and [Formula: see text] be positive integers with[Formula: see text], [Formula: see text] be an integer-valued arithmetic function, and the set [Formula: see text] of [Formula: see text] distinct positive integers be a divisor chain such that [Formula: see text]. We first show that the matrix [Formula: see text] having [Formula: see text] evaluated at the [Formula: see text]th power [Formula: see text] of the greatest common divisor of [Formula: see text] and [Formula: see text] as its [Formula: see text]-entry divides the GCD matrix [Formula: see text] in the ring [Formula: see text] of [Formula: see text] matrices over integers if and only if [Formula: see text] and [Formula: see text] divides [Formula: see text] for any integer [Formula: see text] with [Formula: see text]. Consequently, we show that the matrix [Formula: see text] having [Formula: see text] evaluated at the [Formula: see text]th power [Formula: see text] of the least common multiple of [Formula: see text] and [Formula: see text] as its [Formula: see text]-entry divides the matrix [Formula: see text] in the ring [Formula: see text] if and only if [Formula: see text] and [Formula: see text] divides [Formula: see text] for any integer [Formula: see text] with[Formula: see text]. Finally, we prove that the matrix [Formula: see text] divides the matrix [Formula: see text] in the ring [Formula: see text] if and only if [Formula: see text] and [Formula: see text] for any integer [Formula: see text] with [Formula: see text]. Our results extend and strengthen the theorems of Hong obtained in 2008.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81406439","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-26DOI: 10.1142/s1005386722000360
Dejun Wu, Hui-Shan Zhou
Let [Formula: see text] and [Formula: see text] be rings and [Formula: see text] a [Formula: see text]-bimodule. If [Formula: see text] is flat and [Formula: see text] is finitely generated projective (resp., [Formula: see text] is finitely generated projective and [Formula: see text] 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 [Formula: see text] are given. As applications, it is proved that every Gorenstein AC-projective left [Formula: see text]-module is projective if and only if each Gorenstein AC-projective left [Formula: see text]-module and [Formula: see text]-module is projective, and every Gorenstein AC-injective left [Formula: see text]-module is injective if and only if each Gorenstein AC-injective left [Formula: see text]-module and [Formula: see text]-module is injective. Moreover, Gorenstein AC-projective and AC-injective dimensions over the formal triangular matrix ring [Formula: see text] are studied.
设[公式:见文]和[公式:见文]为环，[公式:见文]为双模。如果[Formula: see text]是平面的，而[Formula: see text]是有限生成的投影(见图1)。，[公式:见文]是有限生成的投影，[公式:见文]是平面的)，那么关卡模块和Gorenstein ac -投影模块的特征(见文)。给出了形式三角矩阵环上的绝对清洁模和Gorenstein ac -内射模[公式:见文]。作为应用，证明了当且仅当每个Gorenstein ac -射影左[公式:见文]-模都是射影，当且仅当每个Gorenstein ac -射影左[公式:见文]-模都是射影，并且每个Gorenstein ac -内射左[公式:见文]-模都是内射，当且仅当每个Gorenstein ac -内射左[公式:见文]-模都是内射。此外，研究了形式三角矩阵环上的Gorenstein ac -射影维数和ac -内射维数[公式:见文]。
{"title":"Gorenstein AC-Projective and AC-Injective Modules over Formal Triangular Matrix Rings","authors":"Dejun Wu, Hui-Shan Zhou","doi":"10.1142/s1005386722000360","DOIUrl":"https://doi.org/10.1142/s1005386722000360","url":null,"abstract":"Let [Formula: see text] and [Formula: see text] be rings and [Formula: see text] a [Formula: see text]-bimodule. If [Formula: see text] is flat and [Formula: see text] is finitely generated projective (resp., [Formula: see text] is finitely generated projective and [Formula: see text] 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 [Formula: see text] are given. As applications, it is proved that every Gorenstein AC-projective left [Formula: see text]-module is projective if and only if each Gorenstein AC-projective left [Formula: see text]-module and [Formula: see text]-module is projective, and every Gorenstein AC-injective left [Formula: see text]-module is injective if and only if each Gorenstein AC-injective left [Formula: see text]-module and [Formula: see text]-module is injective. Moreover, Gorenstein AC-projective and AC-injective dimensions over the formal triangular matrix ring [Formula: see text] are studied.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79624155","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-26DOI: 10.1142/S1005386722000384
Carmelo Cisto
In this paper we introduce a particular semigroup transform [Formula: see text] 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.
{"title":"On Some Numerical Semigroup Transforms","authors":"Carmelo Cisto","doi":"10.1142/S1005386722000384","DOIUrl":"https://doi.org/10.1142/S1005386722000384","url":null,"abstract":"In this paper we introduce a particular semigroup transform [Formula: see text] 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.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85608840","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-30DOI: 10.1142/s1005386722000189
T. Asir, V. Rabikka, H. Su
Let [Formula: see text] be a ring with non-zero identity. The unit graph of [Formula: see text], denoted by [Formula: see text], is an undirected graph with all the elements of [Formula: see text] as vertices and where distinct vertices [Formula: see text], [Formula: see text] are adjacent if and only if [Formula: see text] is a unit of [Formula: see text]. In this paper, we investigate the Wiener index and hyper-Wiener index of [Formula: see text] and explicitly determine their values.
{"title":"On Wiener Index of Unit Graph Associated with a Commutative Ring","authors":"T. Asir, V. Rabikka, H. Su","doi":"10.1142/s1005386722000189","DOIUrl":"https://doi.org/10.1142/s1005386722000189","url":null,"abstract":"Let [Formula: see text] be a ring with non-zero identity. The unit graph of [Formula: see text], denoted by [Formula: see text], is an undirected graph with all the elements of [Formula: see text] as vertices and where distinct vertices [Formula: see text], [Formula: see text] are adjacent if and only if [Formula: see text] is a unit of [Formula: see text]. In this paper, we investigate the Wiener index and hyper-Wiener index of [Formula: see text] and explicitly determine their values.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74034326","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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