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{"title":"关于一个子模块的-Semiannihilator 小子模块和-小子模块的一些性质","authors":"F. Farzalipour, S. Rajaee, P. Ghiasvand","doi":"10.1155/2024/5547197","DOIUrl":null,"url":null,"abstract":"Let <svg height=\"8.8423pt\" style=\"vertical-align:-0.2064009pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 8.28119 8.8423\" width=\"8.28119pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g></svg> be a commutative ring with nonzero identity, <span><svg height=\"9.75571pt\" style=\"vertical-align:-1.11981pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 17.399 9.75571\" width=\"17.399pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-84\"></use></g><g transform=\"matrix(.013,0,0,-0.013,9.768,0)\"></path></g></svg><span></span><svg height=\"9.75571pt\" style=\"vertical-align:-1.11981pt\" version=\"1.1\" viewbox=\"20.9811838 -8.6359 8.332 9.75571\" width=\"8.332pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,21.031,0)\"><use xlink:href=\"#g113-83\"></use></g></svg></span> be a multiplicatively closed subset of <span><svg height=\"8.8423pt\" style=\"vertical-align:-0.2064009pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 8.28119 8.8423\" width=\"8.28119pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-83\"></use></g></svg>,</span> and <svg height=\"8.68572pt\" style=\"vertical-align:-0.0498209pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 12.9526 8.68572\" width=\"12.9526pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g></svg> be a unital <span><svg height=\"8.8423pt\" style=\"vertical-align:-0.2064009pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 8.28119 8.8423\" width=\"8.28119pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-83\"></use></g></svg>-</span>module. In this article, we introduce the concepts of <span><svg height=\"8.8423pt\" style=\"vertical-align:-0.2064009pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 6.25863 8.8423\" width=\"6.25863pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-84\"></use></g></svg>-</span>semiannihilator small submodules and <span><svg height=\"8.8423pt\" style=\"vertical-align:-0.2064009pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 6.25863 8.8423\" width=\"6.25863pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-84\"></use></g></svg>-</span><svg height=\"9.01194pt\" style=\"vertical-align:-0.04981995pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.96212 8.41168 9.01194\" width=\"8.41168pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g></svg>-small submodules as generalizations of <span><svg height=\"8.8423pt\" style=\"vertical-align:-0.2064009pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 6.25863 8.8423\" width=\"6.25863pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-84\"></use></g></svg>-</span>small submodules. We investigate some basic properties of them and give some characterizations of such submodules, especially for (finitely generated faithful) multiplication modules.","PeriodicalId":54214,"journal":{"name":"Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Some Properties of -Semiannihilator Small Submodules and -Small Submodules with respect to a Submodule\",\"authors\":\"F. Farzalipour, S. Rajaee, P. Ghiasvand\",\"doi\":\"10.1155/2024/5547197\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let <svg height=\\\"8.8423pt\\\" style=\\\"vertical-align:-0.2064009pt\\\" version=\\\"1.1\\\" viewbox=\\\"-0.0498162 -8.6359 8.28119 8.8423\\\" width=\\\"8.28119pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,0,0)\\\"></path></g></svg> be a commutative ring with nonzero identity, <span><svg height=\\\"9.75571pt\\\" style=\\\"vertical-align:-1.11981pt\\\" version=\\\"1.1\\\" viewbox=\\\"-0.0498162 -8.6359 17.399 9.75571\\\" width=\\\"17.399pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,0,0)\\\"><use xlink:href=\\\"#g113-84\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,9.768,0)\\\"></path></g></svg><span></span><svg height=\\\"9.75571pt\\\" style=\\\"vertical-align:-1.11981pt\\\" version=\\\"1.1\\\" viewbox=\\\"20.9811838 -8.6359 8.332 9.75571\\\" width=\\\"8.332pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,21.031,0)\\\"><use xlink:href=\\\"#g113-83\\\"></use></g></svg></span> be a multiplicatively closed subset of <span><svg height=\\\"8.8423pt\\\" style=\\\"vertical-align:-0.2064009pt\\\" version=\\\"1.1\\\" viewbox=\\\"-0.0498162 -8.6359 8.28119 8.8423\\\" width=\\\"8.28119pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,0,0)\\\"><use xlink:href=\\\"#g113-83\\\"></use></g></svg>,</span> and <svg height=\\\"8.68572pt\\\" style=\\\"vertical-align:-0.0498209pt\\\" version=\\\"1.1\\\" viewbox=\\\"-0.0498162 -8.6359 12.9526 8.68572\\\" width=\\\"12.9526pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,0,0)\\\"></path></g></svg> be a unital <span><svg height=\\\"8.8423pt\\\" style=\\\"vertical-align:-0.2064009pt\\\" version=\\\"1.1\\\" viewbox=\\\"-0.0498162 -8.6359 8.28119 8.8423\\\" width=\\\"8.28119pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,0,0)\\\"><use xlink:href=\\\"#g113-83\\\"></use></g></svg>-</span>module. In this article, we introduce the concepts of <span><svg height=\\\"8.8423pt\\\" style=\\\"vertical-align:-0.2064009pt\\\" version=\\\"1.1\\\" viewbox=\\\"-0.0498162 -8.6359 6.25863 8.8423\\\" width=\\\"6.25863pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,0,0)\\\"><use xlink:href=\\\"#g113-84\\\"></use></g></svg>-</span>semiannihilator small submodules and <span><svg height=\\\"8.8423pt\\\" style=\\\"vertical-align:-0.2064009pt\\\" version=\\\"1.1\\\" viewbox=\\\"-0.0498162 -8.6359 6.25863 8.8423\\\" width=\\\"6.25863pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,0,0)\\\"><use xlink:href=\\\"#g113-84\\\"></use></g></svg>-</span><svg height=\\\"9.01194pt\\\" style=\\\"vertical-align:-0.04981995pt\\\" version=\\\"1.1\\\" viewbox=\\\"-0.0498162 -8.96212 8.41168 9.01194\\\" width=\\\"8.41168pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,0,0)\\\"></path></g></svg>-small submodules as generalizations of <span><svg height=\\\"8.8423pt\\\" style=\\\"vertical-align:-0.2064009pt\\\" version=\\\"1.1\\\" viewbox=\\\"-0.0498162 -8.6359 6.25863 8.8423\\\" width=\\\"6.25863pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,0,0)\\\"><use xlink:href=\\\"#g113-84\\\"></use></g></svg>-</span>small submodules. We investigate some basic properties of them and give some characterizations of such submodules, especially for (finitely generated faithful) multiplication modules.\",\"PeriodicalId\":54214,\"journal\":{\"name\":\"Journal of Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2024-01-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1155/2024/5547197\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1155/2024/5547197","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
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