A module M is said to belifting if, for anysubmodule N of M, there exists a direct summand X of M contained in N such that N/X is small in M/X. A module M is said to satisfy the finite internal exchange propertyif, for any direct summand X of M and any finite direct sum decomposition M=Lni=1Mi, there exists a direct summand M′i of Mi (i= 1,2, . . . , n) such that M=X⊕(Lni=1M′i). In this paper, we first give characterizations forthe square of a hollow and uniform module to be lifting (extending). In addition, we solve negatively the question "Does any lifting module satisfy the finite internal exchange property?" as an application of this result.
{"title":"On lifting and extending properties on direct sums of hollow uniform modules","authors":"Yoshiharu Shibata","doi":"10.12958/adm1643","DOIUrl":"https://doi.org/10.12958/adm1643","url":null,"abstract":"A module M is said to belifting if, for anysubmodule N of M, there exists a direct summand X of M contained in N such that N/X is small in M/X. A module M is said to satisfy the finite internal exchange propertyif, for any direct summand X of M and any finite direct sum decomposition M=Lni=1Mi, there exists a direct summand M′i of Mi (i= 1,2, . . . , n) such that M=X⊕(Lni=1M′i). In this paper, we first give characterizations forthe square of a hollow and uniform module to be lifting (extending). In addition, we solve negatively the question \"Does any lifting module satisfy the finite internal exchange property?\" as an application of this result.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66418912","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We warmly congratulate N. N. Semko on his 65th birthday and wish him strong health and many successful years of research and teaching.
我们热烈祝贺塞姆科先生65岁生日，并祝他身体健康，科研和教学事业取得成功。
{"title":"Mykola M. Semko (dedicated to the 65th Birthday)","authors":"","doi":"10.12958/adm2062","DOIUrl":"https://doi.org/10.12958/adm2062","url":null,"abstract":"We warmly congratulate N. N. Semko on his 65th birthday and wish him strong health and many successful years of research and teaching.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"54 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66421452","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we show that under some mild assumptions two copies of a metric group can be freely amalgamated over any central subgroup so that the distance between them is sufficiently small.
{"title":"An amalgamation property for metric groups","authors":"Jessica Popowicz, A. Ivanov","doi":"10.12958/adm1557","DOIUrl":"https://doi.org/10.12958/adm1557","url":null,"abstract":"In this paper we show that under some mild assumptions two copies of a metric group can be freely amalgamated over any central subgroup so that the distance between them is sufficiently small.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66418127","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this survey paper the authors specify all the known findings related to the norms of a~group and their generalizations (since 2016 in more details). Special attention is paid to the analysis of their own study of different generalized norms, particularly the norm of non-cyclic subgroups, the norm of Abelian non-cyclic subgroups, the norm of decomposable subgroups and relations between them.
{"title":"Generalized norms of groups: retrospective review and current status","authors":"T. Lukashova, M. Drushlyak","doi":"10.12958/adm1968","DOIUrl":"https://doi.org/10.12958/adm1968","url":null,"abstract":"In this survey paper the authors specify all the known findings related to the norms of a~group and their generalizations (since 2016 in more details). Special attention is paid to the analysis of their own study of different generalized norms, particularly the norm of non-cyclic subgroups, the norm of Abelian non-cyclic subgroups, the norm of decomposable subgroups and relations between them.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66420519","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Reduction Theorem in Leavitt path algebra over a commutative unital ring is very important to prove that the Leavitt path algebra is semiprime if and only if the ring is also semiprime. Any minimal ideal in the semiprime ring and line point will construct a left minimal ideal in the Leavitt path algebra. Vice versa, any left minimal ideal in the semiprime Leavitt path algebra can be found both minimal ideal in the semiprime ring and line point that generate it. The socle of semiprime Leavitt path algebra is constructed by minimal ideals of the semiprime ring and the set of all line points.
{"title":"The socle of Leavitt path algebras over a semiprime ring","authors":"K. Wardati","doi":"10.12958/adm1850","DOIUrl":"https://doi.org/10.12958/adm1850","url":null,"abstract":"The Reduction Theorem in Leavitt path algebra over a commutative unital ring is very important to prove that the Leavitt path algebra is semiprime if and only if the ring is also semiprime. Any minimal ideal in the semiprime ring and line point will construct a left minimal ideal in the Leavitt path algebra. Vice versa, any left minimal ideal in the semiprime Leavitt path algebra can be found both minimal ideal in the semiprime ring and line point that generate it. The socle of semiprime Leavitt path algebra is constructed by minimal ideals of the semiprime ring and the set of all line points.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66420239","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we introduce the concept of (amply) cofinitely ss-supplemented modules as a proper generalization of (amply) ss-supplemented modules, and we provide various properties of these modules. In particular, we prove that arbitrary sum of cofinitely ss-supplemented modules is cofinitely ss-supplemented. Moreover, we show that a ring R is semiperfect and Rad(R)⊆Soc(RR) if and only if every left R-module (amply) cofinitely ss-supplemented.
{"title":"On cofinitely ss-supplemented modules","authors":"B. Türkmen, B. Kılıç","doi":"10.12958/adm1668","DOIUrl":"https://doi.org/10.12958/adm1668","url":null,"abstract":"In this paper, we introduce the concept of (amply) cofinitely ss-supplemented modules as a proper generalization of (amply) ss-supplemented modules, and we provide various properties of these modules. In particular, we prove that arbitrary sum of cofinitely ss-supplemented modules is cofinitely ss-supplemented. Moreover, we show that a ring R is semiperfect and Rad(R)⊆Soc(RR) if and only if every left R-module (amply) cofinitely ss-supplemented.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66419387","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Elementary arguments show that a tree or forest is determined (up to isomorphism) by binary matroids defined using the adjacency matrix.
基本参数表明树或森林是由使用邻接矩阵定义的二元拟阵确定的(直到同构)。
{"title":"Binary matroids that classify forests","authors":"L. Traldi","doi":"10.12958/adm1759","DOIUrl":"https://doi.org/10.12958/adm1759","url":null,"abstract":"Elementary arguments show that a tree or forest is determined (up to isomorphism) by binary matroids defined using the adjacency matrix.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66419739","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abderrahim El Moussaouy, A. R. Moniri Hamzekolaee, M. Ziane
The study of modules by properties of their endomorphisms has long been of interest. In this paper we introduce a proper generalization of that of Hopfian modules, called Jacobson Hopfian modules. A right R-module M is said to be Jacobson Hopfian, if any surjective endomorphism of M has a Jacobson-small kernel. We characterize the rings R for which every finitely generated free R-module is Jacobson Hopfian. We prove that a ring R is semisimple if and only if every R-module is Jacobson Hopfian. Some other properties and characterizations of Jacobson Hopfian modules are also obtained with examples. Further, we prove that the Jacobson Hopfian property is preserved under Morita equivalences.
{"title":"Jacobson Hopfian modules","authors":"Abderrahim El Moussaouy, A. R. Moniri Hamzekolaee, M. Ziane","doi":"10.12958/adm1842","DOIUrl":"https://doi.org/10.12958/adm1842","url":null,"abstract":"The study of modules by properties of their endomorphisms has long been of interest. In this paper we introduce a proper generalization of that of Hopfian modules, called Jacobson Hopfian modules. A right R-module M is said to be Jacobson Hopfian, if any surjective endomorphism of M has a Jacobson-small kernel. We characterize the rings R for which every finitely generated free R-module is Jacobson Hopfian. We prove that a ring R is semisimple if and only if every R-module is Jacobson Hopfian. Some other properties and characterizations of Jacobson Hopfian modules are also obtained with examples. Further, we prove that the Jacobson Hopfian property is preserved under Morita equivalences.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66420039","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, the central and non central codesof semisimple dihedral group algebra FqG, over a finite field Fq, are constructed. Further the distances of these central and non central codes are computed.
{"title":"Central and non central codes of dihedral 2-groups","authors":"Shalini Gupta, P. Rani","doi":"10.12958/adm1569","DOIUrl":"https://doi.org/10.12958/adm1569","url":null,"abstract":"In this paper, the central and non central codesof semisimple dihedral group algebra FqG, over a finite field Fq, are constructed. Further the distances of these central and non central codes are computed.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66418528","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let KG be the modular group algebra of anarbitrary group G over a field K of characteristic p>0. In thispaper we give some improvements of upper Lie nilpotency indext L(KG) of the group algebra KG. It can be seen that if KG is Lie nilpotent, then its lower as well as upper Lie nilpotency index is atleast p+1. In this way the classification of group algebras KG with next upper Lie nilpotency indext L(KG) up to 9p−7 have alreadybeen classified. Furthermore, we give a complete classification ofmodular group algebraKGfor which the upper Lie nilpotency index is 10p−8.
{"title":"A note on modular group algebras with upper Lie nilpotency indices","authors":"Suchi Bhatt, H. Chandra","doi":"10.12958/adm1694","DOIUrl":"https://doi.org/10.12958/adm1694","url":null,"abstract":"Let KG be the modular group algebra of anarbitrary group G over a field K of characteristic p>0. In thispaper we give some improvements of upper Lie nilpotency indext L(KG) of the group algebra KG. It can be seen that if KG is Lie nilpotent, then its lower as well as upper Lie nilpotency index is atleast p+1. In this way the classification of group algebras KG with next upper Lie nilpotency indext L(KG) up to 9p−7 have alreadybeen classified. Furthermore, we give a complete classification ofmodular group algebraKGfor which the upper Lie nilpotency index is 10p−8.","PeriodicalId":44176,"journal":{"name":"Algebra & Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66419441","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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