Pub Date : 2023-08-29DOI: 10.1142/s1005386723000391
Adam Anebri, N. Mahdou, Ünsal Tekir, E. Yıldız
Let [Formula: see text] be a commutative ring with nonzero identity and [Formula: see text] be a positive integer. In this paper, we introduce and investigate a new subclass of [Formula: see text]-[Formula: see text]-absorbing primary ideals, which are called [Formula: see text]-[Formula: see text]-ideals. Let [Formula: see text] be a function, where [Formula: see text] denotes the set of all ideals of [Formula: see text]. A proper ideal [Formula: see text] of [Formula: see text] is called a [Formula: see text]-[Formula: see text]-ideal if [Formula: see text] and [Formula: see text] imply that the product of [Formula: see text] with [Formula: see text] of [Formula: see text] is in [Formula: see text] for all [Formula: see text]. In addition to giving many properties of [Formula: see text]-[Formula: see text]-ideals, we also use the concept of [Formula: see text]-[Formula: see text]-ideals to characterize rings that have only finitely many minimal prime ideals.
{"title":"On ϕ - ( n,N ) -ideals of Commutative Rings","authors":"Adam Anebri, N. Mahdou, Ünsal Tekir, E. Yıldız","doi":"10.1142/s1005386723000391","DOIUrl":"https://doi.org/10.1142/s1005386723000391","url":null,"abstract":"Let [Formula: see text] be a commutative ring with nonzero identity and [Formula: see text] be a positive integer. In this paper, we introduce and investigate a new subclass of [Formula: see text]-[Formula: see text]-absorbing primary ideals, which are called [Formula: see text]-[Formula: see text]-ideals. Let [Formula: see text] be a function, where [Formula: see text] denotes the set of all ideals of [Formula: see text]. A proper ideal [Formula: see text] of [Formula: see text] is called a [Formula: see text]-[Formula: see text]-ideal if [Formula: see text] and [Formula: see text] imply that the product of [Formula: see text] with [Formula: see text] of [Formula: see text] is in [Formula: see text] for all [Formula: see text]. In addition to giving many properties of [Formula: see text]-[Formula: see text]-ideals, we also use the concept of [Formula: see text]-[Formula: see text]-ideals to characterize rings that have only finitely many minimal prime ideals.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84912807","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 : 2023-08-29DOI: 10.1142/s1005386723000408
Shu Li, Jianfeng Wang
A signed graph [Formula: see text] is a graph [Formula: see text] with vertex set [Formula: see text] and edge set [Formula: see text], together with a function [Formula: see text] assigning a positive or negative sign to each edge. In this paper, we present a more elementary proof for the matrix-tree theorem of signed graphs, which is based on the relations between the incidence matrices and the Laplcians of signed graphs. As an application, we also obtain the results of Monfared and Mallik about the matrix-tree theorem of graphs for signless Laplacians.
{"title":"Yet More Elementary Proof of Matrix-Tree Theorem for Signed Graphs","authors":"Shu Li, Jianfeng Wang","doi":"10.1142/s1005386723000408","DOIUrl":"https://doi.org/10.1142/s1005386723000408","url":null,"abstract":"A signed graph [Formula: see text] is a graph [Formula: see text] with vertex set [Formula: see text] and edge set [Formula: see text], together with a function [Formula: see text] assigning a positive or negative sign to each edge. In this paper, we present a more elementary proof for the matrix-tree theorem of signed graphs, which is based on the relations between the incidence matrices and the Laplcians of signed graphs. As an application, we also obtain the results of Monfared and Mallik about the matrix-tree theorem of graphs for signless Laplacians.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89043727","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}
Let [Formula: see text] be a ring with an endomorphism [Formula: see text], [Formula: see text] the free monoid generated by [Formula: see text] with 0 added, and [Formula: see text] a factor of [Formula: see text] obtained by setting certain monomials in [Formula: see text] to 0 such that [Formula: see text] for some [Formula: see text]. Then we can form the non-semiprime skew monoid ring [Formula: see text]. A local ring [Formula: see text] is called bleached if for any [Formula: see text] and any [Formula: see text], the abelian group endomorphisms [Formula: see text] and [Formula: see text] of [Formula: see text] are surjective. Using [Formula: see text], we provide various classes of both bleached and non-bleached local rings. One of the main problems concerning strongly clean rings is to characterize the rings [Formula: see text] for which the matrix ring [Formula: see text] is strongly clean. We investigate the strong cleanness of the full matrix rings over the skew monoid ring [Formula: see text].
{"title":"Strongly Clean Matrix Rings over a Skew Monoid Ring","authors":"Arezou Karimimansoub, Mohammad-Reza (Rafsanjani) Sadeghi","doi":"10.1142/s1005386723000305","DOIUrl":"https://doi.org/10.1142/s1005386723000305","url":null,"abstract":"Let [Formula: see text] be a ring with an endomorphism [Formula: see text], [Formula: see text] the free monoid generated by [Formula: see text] with 0 added, and [Formula: see text] a factor of [Formula: see text] obtained by setting certain monomials in [Formula: see text] to 0 such that [Formula: see text] for some [Formula: see text]. Then we can form the non-semiprime skew monoid ring [Formula: see text]. A local ring [Formula: see text] is called bleached if for any [Formula: see text] and any [Formula: see text], the abelian group endomorphisms [Formula: see text] and [Formula: see text] of [Formula: see text] are surjective. Using [Formula: see text], we provide various classes of both bleached and non-bleached local rings. One of the main problems concerning strongly clean rings is to characterize the rings [Formula: see text] for which the matrix ring [Formula: see text] is strongly clean. We investigate the strong cleanness of the full matrix rings over the skew monoid ring [Formula: see text].","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90698384","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 : 2023-08-29DOI: 10.1142/s100538672300041x
Zhaobing Fan, Shaolong Han, Seok-Jin Kang, Young Rock Kim
Using new combinatorics of Young walls, we give a new construction of the arbitrary level highest weight crystal [Formula: see text] for the quantum affine algebras of types [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. We show that the crystal consisting of reduced Young walls is isomorphic to the crystal [Formula: see text]. Moreover, we provide a new realization of the crystal [Formula: see text] in terms of reduced virtual Young walls and reduced extended Young walls.
{"title":"A New Young Wall Realization of B ( λ ) and B ( ∞ )","authors":"Zhaobing Fan, Shaolong Han, Seok-Jin Kang, Young Rock Kim","doi":"10.1142/s100538672300041x","DOIUrl":"https://doi.org/10.1142/s100538672300041x","url":null,"abstract":"Using new combinatorics of Young walls, we give a new construction of the arbitrary level highest weight crystal [Formula: see text] for the quantum affine algebras of types [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. We show that the crystal consisting of reduced Young walls is isomorphic to the crystal [Formula: see text]. Moreover, we provide a new realization of the crystal [Formula: see text] in terms of reduced virtual Young walls and reduced extended Young walls.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72529412","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 : 2023-08-29DOI: 10.1142/s1005386723000366
Shouqiang Shen, Weijun Liu, Lihua Feng
For a ring [Formula: see text] (not necessarily commutative) with identity, the comaximal graph of [Formula: see text], denoted by [Formula: see text], is a graph whose vertices are all the nonunit elements of [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper we consider a subgraph [Formula: see text] of [Formula: see text] induced by [Formula: see text], where [Formula: see text] is the set of all left-invertible elements of [Formula: see text]. We characterize those rings [Formula: see text] for which [Formula: see text] is a complete graph or a star graph, where [Formula: see text] is the Jacobson radical of [Formula: see text]. We investigate the clique number and the chromatic number of the graph [Formula: see text], and we prove that if every left ideal of [Formula: see text] is symmetric, then this graph is connected and its diameter is at most 3. Moreover, we completely characterize the diameter of [Formula: see text]. We also investigate the properties of [Formula: see text] when [Formula: see text] is a split graph.
对于具有恒等式的环[公式:见文](不一定交换)，[公式:见文]的最大图，用[公式:见文]表示，其顶点是[公式:见文]的所有非单位元素，且两个不同的顶点[公式:见文]和[公式:见文]相邻当且仅当[公式:见文]。本文考虑由[公式:见文]导出的[公式:见文]的一个子图[公式:见文]，其中[公式:见文]是[公式:见文]的所有左可逆元素的集合。我们描述那些[公式:见文]是完全图或星图的环[公式:见文]，其中[公式:见文]是[公式:见文]的Jacobson根。我们研究了图[公式:见文]的团数和色数，并证明了如果[公式:见文]的每一个左理想都是对称的，那么这个图是连通的，并且它的直径不超过3。此外，我们完全描述了[公式:见文本]的直径。我们还研究了当[Formula: see text]是一个分割图时[Formula: see text]的性质。
{"title":"The Comaximal Graphs of Noncommutative Rings","authors":"Shouqiang Shen, Weijun Liu, Lihua Feng","doi":"10.1142/s1005386723000366","DOIUrl":"https://doi.org/10.1142/s1005386723000366","url":null,"abstract":"For a ring [Formula: see text] (not necessarily commutative) with identity, the comaximal graph of [Formula: see text], denoted by [Formula: see text], is a graph whose vertices are all the nonunit elements of [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper we consider a subgraph [Formula: see text] of [Formula: see text] induced by [Formula: see text], where [Formula: see text] is the set of all left-invertible elements of [Formula: see text]. We characterize those rings [Formula: see text] for which [Formula: see text] is a complete graph or a star graph, where [Formula: see text] is the Jacobson radical of [Formula: see text]. We investigate the clique number and the chromatic number of the graph [Formula: see text], and we prove that if every left ideal of [Formula: see text] is symmetric, then this graph is connected and its diameter is at most 3. Moreover, we completely characterize the diameter of [Formula: see text]. We also investigate the properties of [Formula: see text] when [Formula: see text] is a split graph.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81662642","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 : 2023-08-29DOI: 10.1142/s100538672300038x
Liufeng Cao, Xuejun Xia, Libin Li
Let [Formula: see text] be a finite-dimensional pointed rank one Hopf algebra of nilpotent type over a finite group [Formula: see text]. In this paper, we investigate the McKay matrix [Formula: see text] of [Formula: see text] for tensoring with the 2-dimensional indecomposable [Formula: see text]-module [Formula: see text]. It turns out that the characteristic polynomial, eigenvalues and eigenvectors of [Formula: see text] are related to the character table of the finite group [Formula: see text] and a kind of generalized Fibonacci polynomial. Moreover, we construct some eigenvectors of each eigenvalue for [Formula: see text] by using the factorization of the generalized Fibonacci polynomial. As an example, we explicitly compute the characteristic polynomial and eigenvalues of [Formula: see text] and give all eigenvectors of each eigenvalue for [Formula: see text] when [Formula: see text] is a dihedral group of order [Formula: see text].
{"title":"McKay Matrices for Pointed Rank One Hopf Algebras of Nilpotent Type","authors":"Liufeng Cao, Xuejun Xia, Libin Li","doi":"10.1142/s100538672300038x","DOIUrl":"https://doi.org/10.1142/s100538672300038x","url":null,"abstract":"Let [Formula: see text] be a finite-dimensional pointed rank one Hopf algebra of nilpotent type over a finite group [Formula: see text]. In this paper, we investigate the McKay matrix [Formula: see text] of [Formula: see text] for tensoring with the 2-dimensional indecomposable [Formula: see text]-module [Formula: see text]. It turns out that the characteristic polynomial, eigenvalues and eigenvectors of [Formula: see text] are related to the character table of the finite group [Formula: see text] and a kind of generalized Fibonacci polynomial. Moreover, we construct some eigenvectors of each eigenvalue for [Formula: see text] by using the factorization of the generalized Fibonacci polynomial. As an example, we explicitly compute the characteristic polynomial and eigenvalues of [Formula: see text] and give all eigenvectors of each eigenvalue for [Formula: see text] when [Formula: see text] is a dihedral group of order [Formula: see text].","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85804283","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 : 2023-08-29DOI: 10.1142/s1005386723000354
Hongjia Chen, Omer Hassan
In this note, the compatible left-symmetric superalgebra structures on an infinite-dimensional Lie superalgebra with some natural grading conditions are completely determined. The results of earlier work on left-symmetric algebra structures on the Virasoro algebra play an essential role in determining these compatible structures.
{"title":"Left-Symmetric Superalgebra Structures on an Infinite-Dimensional Lie Superalgebra","authors":"Hongjia Chen, Omer Hassan","doi":"10.1142/s1005386723000354","DOIUrl":"https://doi.org/10.1142/s1005386723000354","url":null,"abstract":"In this note, the compatible left-symmetric superalgebra structures on an infinite-dimensional Lie superalgebra with some natural grading conditions are completely determined. The results of earlier work on left-symmetric algebra structures on the Virasoro algebra play an essential role in determining these compatible structures.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83576770","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 : 2023-06-01DOI: 10.1142/s1005386723000251
Mengdie Zhang, Chaowen Zhang
Let [Formula: see text] be a finite-dimensional algebra over the real number field. We prove that the repetitive algebra [Formula: see text] admits the dichotomy property of representation type, i.e., [Formula: see text] is either of discrete representation type or of strongly unbounded type.
{"title":"The Representation Type of Repetitive Algebras over the Real Number Field","authors":"Mengdie Zhang, Chaowen Zhang","doi":"10.1142/s1005386723000251","DOIUrl":"https://doi.org/10.1142/s1005386723000251","url":null,"abstract":"Let [Formula: see text] be a finite-dimensional algebra over the real number field. We prove that the repetitive algebra [Formula: see text] admits the dichotomy property of representation type, i.e., [Formula: see text] is either of discrete representation type or of strongly unbounded type.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79698602","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}
The new concept “derimorphism” generalizing both derivation and homomorphism is defined. When a derimorphism is invertible, its inverse is a Rota–Baxter operator. The general theory of derimorphism is established. The classification of all derimorphisms over an associative unital algebra is obtained. Contrary to the nonexistence of nontrivial positive derivations, it is shown that nontrivial positive derimorphisms do exist over any pair of opposite orderings over [Formula: see text], the lattice-ordered full matrix algebra and upper triangular matrix algebra over a totally ordered field.
{"title":"Derimorphisms over Algebras and Applications","authors":"X. Cao, S.H. Liu, X.S. Lu, Z.J. Ye, Z.R. Yu, Y.H. Zhang","doi":"10.1142/s1005386723000160","DOIUrl":"https://doi.org/10.1142/s1005386723000160","url":null,"abstract":"The new concept “derimorphism” generalizing both derivation and homomorphism is defined. When a derimorphism is invertible, its inverse is a Rota–Baxter operator. The general theory of derimorphism is established. The classification of all derimorphisms over an associative unital algebra is obtained. Contrary to the nonexistence of nontrivial positive derivations, it is shown that nontrivial positive derimorphisms do exist over any pair of opposite orderings over [Formula: see text], the lattice-ordered full matrix algebra and upper triangular matrix algebra over a totally ordered field.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78602062","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 : 2023-06-01DOI: 10.1142/s1005386723000159
Wei Bai, Wende Liu
Let [Formula: see text] be a simple modular Lie superealgebra of Witt type or special type over an algebraically closed field of characteristic [Formula: see text]. In this paper, all the superbiderivations of [Formula: see text] are studied. By means of weight space decompositions with respect to a suitable torus and the standard [Formula: see text]-grading structures of [Formula: see text], we show that the supersymmetric superbiderivations of [Formula: see text] are zero. Generalizing a result on the skewsymmetric biderivation of Lie algebras to the super case, we find that all the superbiderivations of [Formula: see text] are inner. As applications, the linear supercommuting maps and the supercommutative post-Lie superalgebra structures on [Formula: see text] are described.
{"title":"Superbiderivations of Simple Modular Lie Superalgebras of Witt Type and Special Type","authors":"Wei Bai, Wende Liu","doi":"10.1142/s1005386723000159","DOIUrl":"https://doi.org/10.1142/s1005386723000159","url":null,"abstract":"Let [Formula: see text] be a simple modular Lie superealgebra of Witt type or special type over an algebraically closed field of characteristic [Formula: see text]. In this paper, all the superbiderivations of [Formula: see text] are studied. By means of weight space decompositions with respect to a suitable torus and the standard [Formula: see text]-grading structures of [Formula: see text], we show that the supersymmetric superbiderivations of [Formula: see text] are zero. Generalizing a result on the skewsymmetric biderivation of Lie algebras to the super case, we find that all the superbiderivations of [Formula: see text] are inner. As applications, the linear supercommuting maps and the supercommutative post-Lie superalgebra structures on [Formula: see text] are described.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79049757","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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