{"title":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M1\">\n <mi mathvariant=\"script\">A</mi>\n </math>-D3 Modules and <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M2\">\n <mi mathvariant=\"script\">A</mi>\n </math>-D4 Modules","authors":"Zhanmin Zhu","doi":"10.1155/2023/4148088","DOIUrl":null,"url":null,"abstract":"<jats:p>Let <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M3\">\n <mi mathvariant=\"script\">A</mi>\n </math>\n </jats:inline-formula> be a class of some right <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M4\">\n <mi>R</mi>\n </math>\n </jats:inline-formula>-modules that is closed under isomorphisms, and let <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M5\">\n <mi>M</mi>\n </math>\n </jats:inline-formula> be a right <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M6\">\n <mi>R</mi>\n </math>\n </jats:inline-formula>-module. Then <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M7\">\n <mi>M</mi>\n </math>\n </jats:inline-formula> is called <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M8\">\n <mi mathvariant=\"script\">A</mi>\n </math>\n </jats:inline-formula>-D3 if, whenever <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M9\">\n <mi>N</mi>\n </math>\n </jats:inline-formula> and <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M10\">\n <mi>K</mi>\n </math>\n </jats:inline-formula> are direct summands of <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M11\">\n <mi>M</mi>\n </math>\n </jats:inline-formula> with <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M12\">\n <mi>M</mi>\n <mo>=</mo>\n <mi>N</mi>\n <mo>+</mo>\n <mi>K</mi>\n </math>\n </jats:inline-formula> and <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M13\">\n <mi>M</mi>\n <mo>/</mo>\n <mi>K</mi>\n <mo>∈</mo>\n <mi mathvariant=\"script\">A</mi>\n </math>\n </jats:inline-formula>, then <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M14\">\n <mi>N</mi>\n <mo>∩</mo>\n <mi>K</mi>\n </math>\n </jats:inline-formula> is also a direct summand of <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M15\">\n <mi>M</mi>\n </math>\n </jats:inline-formula>; <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M16\">\n <mi>M</mi>\n </math>\n </jats:inline-formula> is called an <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M17\">\n <mi mathvariant=\"script\">A</mi>\n </math>\n </jats:inline-formula>-D4 module, if whenever <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M18\">\n <mi>M</mi>\n <mo>=</mo>\n <mi>B</mi>\n <mo>⊕</mo>\n <mi>A</mi>\n </math>\n </jats:inline-formula> where <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M19\">\n <mi>B</mi>\n </math>\n </jats:inline-formula> and <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M20\">\n <mi>A</mi>\n </math>\n </jats:inline-formula> are submodules of <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M21\">\n <mi>M</mi>\n </math>\n </jats:inline-formula> and <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M22\">\n <mi>A</mi>\n <mo>∈</mo>\n <mi mathvariant=\"script\">A</mi>\n </math>\n </jats:inline-formula>, then every epimorphism <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M23\">\n <mi>f</mi>\n <mo>:</mo>\n <mi>B</mi>\n <mo>⟶</mo>\n <mi>A</mi>\n </math>\n </jats:inline-formula> splits. Several characterizations and properties of these classes of modules are investigated. As applications, some new characterizations of semisimple Artinian rings, quasi-Frobenius rings, von Neumann regular rings, semiregular rings, perfect rings, semiperfect rings, hereditary rings, semihereditary rings, and PP rings are given.</jats:p>","PeriodicalId":43667,"journal":{"name":"Muenster Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2023-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Muenster Journal of Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1155/2023/4148088","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let be a class of some right -modules that is closed under isomorphisms, and let be a right -module. Then is called -D3 if, whenever and are direct summands of with and , then is also a direct summand of ; is called an -D4 module, if whenever where and are submodules of and , then every epimorphism splits. Several characterizations and properties of these classes of modules are investigated. As applications, some new characterizations of semisimple Artinian rings, quasi-Frobenius rings, von Neumann regular rings, semiregular rings, perfect rings, semiperfect rings, hereditary rings, semihereditary rings, and PP rings are given.