{"title":"当(对偶-)Baer模是(协-)素模的直和时","authors":"¨¨—˚¨¯ ¸¯˚—˛˝˝¯, ¯¨¯˚¨¯ ¨˙´¯¨, N. Ghaedan","doi":"10.33048/semi.2021.18.057","DOIUrl":null,"url":null,"abstract":"Since 2004, Baer modules have been considered by many authors as a generalization of the Baer rings. A module $M_R$ is called Baer if every intersection of the kernels of endomorphisms on $M_R$ is a direct summand of $M_R$. It is known that commutative Baer rings are reduced. We prove that if a Baer module M is a direct sum of prime modules, then every direct summand of M is retractable. The converse is true whenever the triangulating dimension of $M$ is finite (e.g. if the uniform dimension of M is finite). Dually, if every direct summand of a dual-Baer module M is co-retractable, then it is a direct sum of co-prime modules and the converse is true whenever the sum is finite or M is a max-module. Among other applications, we show that if R is a commutative hereditary Noetherian ring then a finitely generated R-module is Baer iff it is projective or semisimple. Also, over a ring Morita equivalent to a perfect duo ring, all dual-Baer modules are semisimple.","PeriodicalId":45858,"journal":{"name":"Siberian Electronic Mathematical Reports-Sibirskie Elektronnye Matematicheskie Izvestiya","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2021-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"When a (dual-)Baer module is a direct sum of (co-)prime modules\",\"authors\":\"¨¨—˚¨¯ ¸¯˚—˛˝˝¯, ¯¨¯˚¨¯ ¨˙´¯¨, N. Ghaedan\",\"doi\":\"10.33048/semi.2021.18.057\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Since 2004, Baer modules have been considered by many authors as a generalization of the Baer rings. A module $M_R$ is called Baer if every intersection of the kernels of endomorphisms on $M_R$ is a direct summand of $M_R$. It is known that commutative Baer rings are reduced. We prove that if a Baer module M is a direct sum of prime modules, then every direct summand of M is retractable. The converse is true whenever the triangulating dimension of $M$ is finite (e.g. if the uniform dimension of M is finite). Dually, if every direct summand of a dual-Baer module M is co-retractable, then it is a direct sum of co-prime modules and the converse is true whenever the sum is finite or M is a max-module. Among other applications, we show that if R is a commutative hereditary Noetherian ring then a finitely generated R-module is Baer iff it is projective or semisimple. Also, over a ring Morita equivalent to a perfect duo ring, all dual-Baer modules are semisimple.\",\"PeriodicalId\":45858,\"journal\":{\"name\":\"Siberian Electronic Mathematical Reports-Sibirskie Elektronnye Matematicheskie Izvestiya\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2021-07-06\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Siberian Electronic Mathematical Reports-Sibirskie Elektronnye Matematicheskie Izvestiya\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.33048/semi.2021.18.057\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Siberian Electronic Mathematical Reports-Sibirskie Elektronnye Matematicheskie Izvestiya","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.33048/semi.2021.18.057","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
When a (dual-)Baer module is a direct sum of (co-)prime modules
Since 2004, Baer modules have been considered by many authors as a generalization of the Baer rings. A module $M_R$ is called Baer if every intersection of the kernels of endomorphisms on $M_R$ is a direct summand of $M_R$. It is known that commutative Baer rings are reduced. We prove that if a Baer module M is a direct sum of prime modules, then every direct summand of M is retractable. The converse is true whenever the triangulating dimension of $M$ is finite (e.g. if the uniform dimension of M is finite). Dually, if every direct summand of a dual-Baer module M is co-retractable, then it is a direct sum of co-prime modules and the converse is true whenever the sum is finite or M is a max-module. Among other applications, we show that if R is a commutative hereditary Noetherian ring then a finitely generated R-module is Baer iff it is projective or semisimple. Also, over a ring Morita equivalent to a perfect duo ring, all dual-Baer modules are semisimple.