Pub Date : 2023-11-13DOI: 10.1142/s0219498825501634
Hannah Fox, Agastya Goel, Sophia Liao
A subset $S$ of an integral domain is called a semidomain if the pairs $(S,+)$ and $(Ssetminus{0}, cdot)$ are commutative and cancellative semigroups with identities. The multiplication of $S$ extends to the group of differences $mathscr{G}(S)$, turning $mathscr{G}(S)$ into an integral domain. In this paper, we study the arithmetic of semisubtractive semidomains (i.e., semidomains $S$ for which either $s in S$ or $-s in S$ for every $s in mathscr{G}(S)$). Specifically, we provide necessary and sufficient conditions for a semisubtractive semidomain to be atomic, to satisfy the ascending chain condition on principals ideals, to be a bounded factorization semidomain, and to be a finite factorization semidomain, which are subsequent relaxations of the property of having unique factorizations. In addition, we present a characterization of factorial and half-factorial semisubtractive semidomains. Throughout the article, we present examples to provide insight into the arithmetic aspects of semisubtractive semidomains.
如果$(S,+)$和$(Ssetminus{0}, cdot)$这两对半群是交换半群和可取消半群，那么一个积分域的子集$S$被称为半域。$S$ 的乘法扩展到差集 $mathscr{G}(S)$，把 $mathscr{G}(S)$ 变成了一个积分域。在本文中，我们将研究半减半域的算术（即对于每一个 $s in S$ 或 $-s in S$ 的半域 $S$）。具体来说，我们提供了半减法半域成为原子半域、满足主ideals的升链条件、成为有界因式分解半域以及成为有限因式分解半域的必要条件和充分条件，这些条件是对具有唯一因式分解这一性质的后续放宽。此外，我们还介绍了因式半域和半因式半域的特征。在整篇文章中，我们举例说明了半减半域的算术方面。
{"title":"Arithmetic of Semisubtractive Semidomains","authors":"Hannah Fox, Agastya Goel, Sophia Liao","doi":"10.1142/s0219498825501634","DOIUrl":"https://doi.org/10.1142/s0219498825501634","url":null,"abstract":"A subset $S$ of an integral domain is called a semidomain if the pairs $(S,+)$ and $(Ssetminus{0}, cdot)$ are commutative and cancellative semigroups with identities. The multiplication of $S$ extends to the group of differences $mathscr{G}(S)$, turning $mathscr{G}(S)$ into an integral domain. In this paper, we study the arithmetic of semisubtractive semidomains (i.e., semidomains $S$ for which either $s in S$ or $-s in S$ for every $s in mathscr{G}(S)$). Specifically, we provide necessary and sufficient conditions for a semisubtractive semidomain to be atomic, to satisfy the ascending chain condition on principals ideals, to be a bounded factorization semidomain, and to be a finite factorization semidomain, which are subsequent relaxations of the property of having unique factorizations. In addition, we present a characterization of factorial and half-factorial semisubtractive semidomains. Throughout the article, we present examples to provide insight into the arithmetic aspects of semisubtractive semidomains.","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"GE-23 4","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139278480","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-08DOI: 10.1142/s0219498825501014
Chongxia Lu, Qingguo Li
In this paper, we discuss the properties of the hull-kernel topology and the inverse topology on the set [Formula: see text] of minimal prime elements of a continuous frame [Formula: see text]. We prove that the set [Formula: see text] endowed with the hull-kernel topology and the inverse topology, respectively, is a [Formula: see text]-space. Moreover, we obtain that there is a bijective correspondence between the set [Formula: see text] of all maximal Scott open filters of [Formula: see text] and the set [Formula: see text] when [Formula: see text] is a stably continuous frame. By considering the bijective correspondence between the sets [Formula: see text] and [Formula: see text], we propose some sufficient conditions for the topological spaces [Formula: see text] endowed with the hull-kernel topology and the inverse topology, respectively, to be sober, Hausdorff, compact, extremely disconnected and zero-dimensional spaces, respectively, when [Formula: see text] is a stably continuous frame.
本文讨论了连续框架最小素元集[公式:见文]上的壳核拓扑和逆拓扑的性质。我们证明了分别具有壳核拓扑和逆拓扑的集合[公式:见文]是一个[公式:见文]空间。并且，我们得到了当[Formula: see text]为稳定连续帧时，[Formula: see text]的所有极大Scott开放滤波器的集合[Formula: see text]与集合[Formula: see text]之间存在双射对应。通过考虑集合[公式:见文]和集合[公式:见文]之间的双射对应关系，给出了当[公式:见文]是稳定连续坐标系时，具有壳核拓扑和逆拓扑的拓扑空间[公式:见文]分别为清醒空间、Hausdorff空间、紧空间、极度不连通空间和零维空间的充分条件。
{"title":"Two topologies on the set of minimal prime elements of a continuous frame","authors":"Chongxia Lu, Qingguo Li","doi":"10.1142/s0219498825501014","DOIUrl":"https://doi.org/10.1142/s0219498825501014","url":null,"abstract":"In this paper, we discuss the properties of the hull-kernel topology and the inverse topology on the set [Formula: see text] of minimal prime elements of a continuous frame [Formula: see text]. We prove that the set [Formula: see text] endowed with the hull-kernel topology and the inverse topology, respectively, is a [Formula: see text]-space. Moreover, we obtain that there is a bijective correspondence between the set [Formula: see text] of all maximal Scott open filters of [Formula: see text] and the set [Formula: see text] when [Formula: see text] is a stably continuous frame. By considering the bijective correspondence between the sets [Formula: see text] and [Formula: see text], we propose some sufficient conditions for the topological spaces [Formula: see text] endowed with the hull-kernel topology and the inverse topology, respectively, to be sober, Hausdorff, compact, extremely disconnected and zero-dimensional spaces, respectively, when [Formula: see text] is a stably continuous frame.","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":" 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135293388","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-08DOI: 10.1142/s0219498825500732
Ahmad Khojali, Naser Zamani, Soodabeh Azimi
Let [Formula: see text] be a [Formula: see text]-graded ring and let [Formula: see text] be the category of [Formula: see text]-graded [Formula: see text]-modules and homogeneous homomorphisms. In this paper, we define and study some objects in this category. More precisely, we introduce the concepts of graded duo (weak and strong graded duo) modules and give some sources and an example for these types of modules. It is seen that, under some condition, graded duo property is a local property in this category. When the ring [Formula: see text] is a discrete graded valuation ring with unique [Formula: see text]maximal ideal [Formula: see text], we see that these three types of graded (duo) modules are identical and give an explicit characterization of them, so that any graded duo modules over such a ring is of the form [Formula: see text] or [Formula: see text] for some positive integer [Formula: see text] and some integers [Formula: see text]. The same task is done whenever [Formula: see text] is a graded Dedekind domain. Finally, by an example, that provides a wide variety of strong graded duo modules, it was shown that the given characterizations do not hold valid if the ground ring is not Dedekind.
{"title":"On some graded objects in graded module category","authors":"Ahmad Khojali, Naser Zamani, Soodabeh Azimi","doi":"10.1142/s0219498825500732","DOIUrl":"https://doi.org/10.1142/s0219498825500732","url":null,"abstract":"Let [Formula: see text] be a [Formula: see text]-graded ring and let [Formula: see text] be the category of [Formula: see text]-graded [Formula: see text]-modules and homogeneous homomorphisms. In this paper, we define and study some objects in this category. More precisely, we introduce the concepts of graded duo (weak and strong graded duo) modules and give some sources and an example for these types of modules. It is seen that, under some condition, graded duo property is a local property in this category. When the ring [Formula: see text] is a discrete graded valuation ring with unique [Formula: see text]maximal ideal [Formula: see text], we see that these three types of graded (duo) modules are identical and give an explicit characterization of them, so that any graded duo modules over such a ring is of the form [Formula: see text] or [Formula: see text] for some positive integer [Formula: see text] and some integers [Formula: see text]. The same task is done whenever [Formula: see text] is a graded Dedekind domain. Finally, by an example, that provides a wide variety of strong graded duo modules, it was shown that the given characterizations do not hold valid if the ground ring is not Dedekind.","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":" 26","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135293664","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-08DOI: 10.1142/s0219498825500781
Kostiantyn Iusenko, John William MacQuarrie
We give a self-contained introduction to the wonderfully well-behaved class of pseudocompact algebras, focusing on the foundational classes of semisimple and separable algebras. We give characterizations of such algebras analogous to those for finite dimensional algebras. We give a self-contained proof of the Wedderburn–Malcev Theorem for pseudocompact algebras.
{"title":"Semisimplicity and separability for pseudocompact algebras","authors":"Kostiantyn Iusenko, John William MacQuarrie","doi":"10.1142/s0219498825500781","DOIUrl":"https://doi.org/10.1142/s0219498825500781","url":null,"abstract":"We give a self-contained introduction to the wonderfully well-behaved class of pseudocompact algebras, focusing on the foundational classes of semisimple and separable algebras. We give characterizations of such algebras analogous to those for finite dimensional algebras. We give a self-contained proof of the Wedderburn–Malcev Theorem for pseudocompact algebras.","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":" 13","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135293673","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-08DOI: 10.1142/s0219498825500914
Itamar Stein
We study the generalized right ample identity, introduced by the author in a previous paper. Let [Formula: see text] be a reduced [Formula: see text]-Fountain semigroup which satisfies the congruence condition. We can associate with [Formula: see text] a small category [Formula: see text] whose set of objects is identified with the set [Formula: see text] of idempotents and its morphisms correspond to elements of [Formula: see text]. We prove that [Formula: see text] satisfies the generalized right ample identity if and only if every element of [Formula: see text] induces a homomorphism of left [Formula: see text]-actions between certain classes of generalized Green’s relations. In this case, we interpret the associated category [Formula: see text] as a discrete form of a Peirce decomposition of the semigroup algebra. We also give some natural examples of semigroups satisfying this identity.
{"title":"Algebras of Reduced <i>E</i>-Fountain Semigroups and the Generalized Ample Identity II","authors":"Itamar Stein","doi":"10.1142/s0219498825500914","DOIUrl":"https://doi.org/10.1142/s0219498825500914","url":null,"abstract":"We study the generalized right ample identity, introduced by the author in a previous paper. Let [Formula: see text] be a reduced [Formula: see text]-Fountain semigroup which satisfies the congruence condition. We can associate with [Formula: see text] a small category [Formula: see text] whose set of objects is identified with the set [Formula: see text] of idempotents and its morphisms correspond to elements of [Formula: see text]. We prove that [Formula: see text] satisfies the generalized right ample identity if and only if every element of [Formula: see text] induces a homomorphism of left [Formula: see text]-actions between certain classes of generalized Green’s relations. In this case, we interpret the associated category [Formula: see text] as a discrete form of a Peirce decomposition of the semigroup algebra. We also give some natural examples of semigroups satisfying this identity.","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":" 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135293562","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-03DOI: 10.1142/s021949882550135x
M. Benelmekki, S. El Baghdadi, S. Najib
{"title":"Factorization in Group Algebras","authors":"M. Benelmekki, S. El Baghdadi, S. Najib","doi":"10.1142/s021949882550135x","DOIUrl":"https://doi.org/10.1142/s021949882550135x","url":null,"abstract":"","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135868022","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-03DOI: 10.1142/s0219498825501348
Septimiu Crivei, Robert Pop
{"title":"Projectivity and Subprojectivity Domains in Exact Categories","authors":"Septimiu Crivei, Robert Pop","doi":"10.1142/s0219498825501348","DOIUrl":"https://doi.org/10.1142/s0219498825501348","url":null,"abstract":"","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"7 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135869136","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-03DOI: 10.1142/s0219498825501038
Fang Du, Hao Wang
For an [Formula: see text]-local vertex operator algebra [Formula: see text], where [Formula: see text] is a simple vertex operator algebra, and [Formula: see text] is the group algebra of a finite subgroup of [Formula: see text], Frobenius reciprocity is investigated. We give an explicit construction and classification of admissible [Formula: see text]-modules in terms of admissible [Formula: see text]-modules. We also give a complete set of irreducible inequivalent admissible [Formula: see text]-modules.
{"title":"Classification of irreducible admissible modules for <i>S</i>-local vertex operator algebra <i>V</i>#<i>H</i>","authors":"Fang Du, Hao Wang","doi":"10.1142/s0219498825501038","DOIUrl":"https://doi.org/10.1142/s0219498825501038","url":null,"abstract":"For an [Formula: see text]-local vertex operator algebra [Formula: see text], where [Formula: see text] is a simple vertex operator algebra, and [Formula: see text] is the group algebra of a finite subgroup of [Formula: see text], Frobenius reciprocity is investigated. We give an explicit construction and classification of admissible [Formula: see text]-modules in terms of admissible [Formula: see text]-modules. We also give a complete set of irreducible inequivalent admissible [Formula: see text]-modules.","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"177 S428","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135775281","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-03DOI: 10.1142/s0219498825500884
M. H. Bien, P. T. Nhan, N. H. T. Nhat
Let [Formula: see text] be a local ring with maximal ideal [Formula: see text], let [Formula: see text] be a natural number greater than [Formula: see text] and let [Formula: see text] be a matrix in the general linear group [Formula: see text] of degree [Formula: see text] over [Formula: see text]. We firstly show that if the matrix [Formula: see text] is nonscalar in [Formula: see text] and [Formula: see text] are invertible elements in [Formula: see text], then there exists an invertible element [Formula: see text] such that [Formula: see text] is similar to the product [Formula: see text] in which [Formula: see text] is a lower uni-triangular matrix and [Formula: see text] is an upper triangular matrix whose diagonal entries are [Formula: see text]. We then present some applications of this factorization to find decompositions of matrices in [Formula: see text] into product of commutators and involutions.
{"title":"Some Certain Decompositions of Matrices Over Local Rings","authors":"M. H. Bien, P. T. Nhan, N. H. T. Nhat","doi":"10.1142/s0219498825500884","DOIUrl":"https://doi.org/10.1142/s0219498825500884","url":null,"abstract":"Let [Formula: see text] be a local ring with maximal ideal [Formula: see text], let [Formula: see text] be a natural number greater than [Formula: see text] and let [Formula: see text] be a matrix in the general linear group [Formula: see text] of degree [Formula: see text] over [Formula: see text]. We firstly show that if the matrix [Formula: see text] is nonscalar in [Formula: see text] and [Formula: see text] are invertible elements in [Formula: see text], then there exists an invertible element [Formula: see text] such that [Formula: see text] is similar to the product [Formula: see text] in which [Formula: see text] is a lower uni-triangular matrix and [Formula: see text] is an upper triangular matrix whose diagonal entries are [Formula: see text]. We then present some applications of this factorization to find decompositions of matrices in [Formula: see text] into product of commutators and involutions.","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"178 S440","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135775420","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-03DOI: 10.1142/s0219498825500963
J. Prabu, J. Mahalakshmi, S. Santhakumar
In this paper, we constructed a class of [Formula: see text]-weight linear codes over [Formula: see text] under the homogeneous weight metric by their generator matrices, where [Formula: see text] and [Formula: see text] The Gray images of some class of these codes over [Formula: see text] are [Formula: see text]-ary nonlinear codes, which have the same weight distributions as that of the two-weight [Formula: see text]-ary linear codes of type SU1 in the sense of [R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bull. London Math. Soc. 18(2) (1986) 97–122]. Also, we obtained the minimum distance of the dual codes of the constructed codes. Further, we discussed some optimal linear codes over [Formula: see text] with respect to Plotkin-type bound from the constructed codes when [Formula: see text] Furthermore, we investigated the applications in strongly regular graphs and secret sharing schemes.
{"title":"A class of <i>t</i>-weight codes and its applications","authors":"J. Prabu, J. Mahalakshmi, S. Santhakumar","doi":"10.1142/s0219498825500963","DOIUrl":"https://doi.org/10.1142/s0219498825500963","url":null,"abstract":"In this paper, we constructed a class of [Formula: see text]-weight linear codes over [Formula: see text] under the homogeneous weight metric by their generator matrices, where [Formula: see text] and [Formula: see text] The Gray images of some class of these codes over [Formula: see text] are [Formula: see text]-ary nonlinear codes, which have the same weight distributions as that of the two-weight [Formula: see text]-ary linear codes of type SU1 in the sense of [R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bull. London Math. Soc. 18(2) (1986) 97–122]. Also, we obtained the minimum distance of the dual codes of the constructed codes. Further, we discussed some optimal linear codes over [Formula: see text] with respect to Plotkin-type bound from the constructed codes when [Formula: see text] Furthermore, we investigated the applications in strongly regular graphs and secret sharing schemes.","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"178 S439","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135775421","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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