{"title":"关于分级模类中的若干分级对象","authors":"Ahmad Khojali, Naser Zamani, Soodabeh Azimi","doi":"10.1142/s0219498825500732","DOIUrl":null,"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":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On some graded objects in graded module category\",\"authors\":\"Ahmad Khojali, Naser Zamani, Soodabeh Azimi\",\"doi\":\"10.1142/s0219498825500732\",\"DOIUrl\":null,\"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\":null,\"pages\":null},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2023-11-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Algebra and Its Applications\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1142/s0219498825500732\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Algebra and Its Applications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1142/s0219498825500732","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
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.
期刊介绍:
The Journal of Algebra and Its Applications will publish papers both on theoretical and on applied aspects of Algebra. There is special interest in papers that point out innovative links between areas of Algebra and fields of application. As the field of Algebra continues to experience tremendous growth and diversification, we intend to provide the mathematical community with a central source for information on both the theoretical and the applied aspects of the discipline. While the journal will be primarily devoted to the publication of original research, extraordinary expository articles that encourage communication between algebraists and experts on areas of application as well as those presenting the state of the art on a given algebraic sub-discipline will be considered.