Sufficient and Necessary Conditions of A Module to be the Unique Factorization Modules
An integral domain D is called a unique factorization domain (UFD) if the following conditions are satisfied: (1) Every element of D that is neither 0 nor a unit can be factored into a product of a finite number of irreducibles, and (2) if p1..., pr and qx..., qs are two factorizations of the same e...
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AIP Conference Proceedings
2022
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| author | Prabhadika, I Putu Yudi Wahyuni, Sri |
| author_facet | Prabhadika, I Putu Yudi Wahyuni, Sri |
| author_sort | Prabhadika, I Putu Yudi |
| collection | UGM |
| description | An integral domain D is called a unique factorization domain (UFD) if the following conditions are satisfied: (1) Every element of D that is neither 0 nor a unit can be factored into a product of a finite number of irreducibles, and (2) if p1..., pr and qx..., qs are two factorizations of the same element of D into irreducibles, then r = s and qj can be renumbered so that pi and qi are associates. It has been known the following equivalent conditions: Let D be an integral domain, the following are equivalent: (i) D is a LTFD, (ii) D is a GCD domain satisfying the ascending chain condition on principal ideals, and (iii) D satisfies the ascending chain condition on principal ideals and every irreducible element of D is a prime element of D. The concept and the equivalent conditions on LTFD, motivate some studies that may apply the concept of factorization to modules in order to obtain a definition of a unique factorization module (UFM). First, the concept of irreducible elements is given in the module which will play an important role in defining the UFM. The definition of primitive elements, pure submodules, least common multiple and greatest common divisor in a module is also given. Next, the definition and characterization of a UFM will be presented. The results of this study is providing the sufficient and necessary conditions of a module to be UFM. © 2022 American Institute of Physics Inc.. All rights reserved |
| first_indexed | 2024-03-14T00:10:21Z |
| format | Other |
| id | oai:generic.eprints.org:284464 |
| institution | Universiti Gadjah Mada |
| last_indexed | 2024-03-14T00:10:21Z |
| publishDate | 2022 |
| publisher | AIP Conference Proceedings |
| record_format | dspace |
| spelling | oai:generic.eprints.org:2844642023-12-21T08:47:00Z https://repository.ugm.ac.id/284464/ Sufficient and Necessary Conditions of A Module to be the Unique Factorization Modules Prabhadika, I Putu Yudi Wahyuni, Sri Pure Mathematics An integral domain D is called a unique factorization domain (UFD) if the following conditions are satisfied: (1) Every element of D that is neither 0 nor a unit can be factored into a product of a finite number of irreducibles, and (2) if p1..., pr and qx..., qs are two factorizations of the same element of D into irreducibles, then r = s and qj can be renumbered so that pi and qi are associates. It has been known the following equivalent conditions: Let D be an integral domain, the following are equivalent: (i) D is a LTFD, (ii) D is a GCD domain satisfying the ascending chain condition on principal ideals, and (iii) D satisfies the ascending chain condition on principal ideals and every irreducible element of D is a prime element of D. The concept and the equivalent conditions on LTFD, motivate some studies that may apply the concept of factorization to modules in order to obtain a definition of a unique factorization module (UFM). First, the concept of irreducible elements is given in the module which will play an important role in defining the UFM. The definition of primitive elements, pure submodules, least common multiple and greatest common divisor in a module is also given. Next, the definition and characterization of a UFM will be presented. The results of this study is providing the sufficient and necessary conditions of a module to be UFM. © 2022 American Institute of Physics Inc.. All rights reserved AIP Conference Proceedings 2022 Other NonPeerReviewed Prabhadika, I Putu Yudi and Wahyuni, Sri (2022) Sufficient and Necessary Conditions of A Module to be the Unique Factorization Modules. AIP Conference Proceedings. https://pubs.aip.org/aip/acp/article-abstract/2639/1/020003/2830964/Sufficient-and-necessary-conditions-of-a-module-to?redirectedFrom=fulltext 10.1063/5.0110885 |
| spellingShingle | Pure Mathematics Prabhadika, I Putu Yudi Wahyuni, Sri Sufficient and Necessary Conditions of A Module to be the Unique Factorization Modules |
| title | Sufficient and Necessary Conditions of A Module to be the Unique Factorization Modules |
| title_full | Sufficient and Necessary Conditions of A Module to be the Unique Factorization Modules |
| title_fullStr | Sufficient and Necessary Conditions of A Module to be the Unique Factorization Modules |
| title_full_unstemmed | Sufficient and Necessary Conditions of A Module to be the Unique Factorization Modules |
| title_short | Sufficient and Necessary Conditions of A Module to be the Unique Factorization Modules |
| title_sort | sufficient and necessary conditions of a module to be the unique factorization modules |
| topic | Pure Mathematics |
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