{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T13:09:00Z","timestamp":1753880940687,"version":"3.41.2"},"reference-count":9,"publisher":"World Scientific Pub Co Pte Ltd","issue":"04","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Discrete Math. Algorithm. Appl."],"published-print":{"date-parts":[[2023,5]]},"abstract":"<jats:p> In this paper, we associate a simple graph to a module over a commutative ring and call it extended graph, that adjacent condition depends on the properties of annihilator submodules. We determine the structure of modules, when the associated graph is star or complete. Moreover, we look for relationships between the extended graph and the zero-divisor graph of modules. <\/jats:p>","DOI":"10.1142\/s1793830922501051","type":"journal-article","created":{"date-parts":[[2022,4,28]],"date-time":"2022-04-28T09:43:54Z","timestamp":1651139034000},"source":"Crossref","is-referenced-by-count":0,"title":["The extended graph of a module over a commutative ring"],"prefix":"10.1142","volume":"15","author":[{"given":"S.","family":"Babaei","sequence":"first","affiliation":[{"name":"Department of Mathematics, Imam Khomeini International University, Postal Code 34149-1-6818 Qazvin, Iran"}]},{"given":"E.","family":"Sengelen Sevim","sequence":"additional","affiliation":[{"name":"Eski Silahtara\u011fa Elektrik Santrali Kazim Karabekir, Istanbul Bilgi University, Cad. No. 2\/1334060, Ey\u00fcp, Istanbul, Turkey"}]}],"member":"219","published-online":{"date-parts":[[2022,5,24]]},"reference":[{"key":"S1793830922501051BIB001","doi-asserted-by":"crossref","first-page":"434","DOI":"10.1006\/jabr.1998.7840","volume":"217","author":"Anderson D. F.","year":"1999","journal-title":"J. Algebra"},{"key":"S1793830922501051BIB002","doi-asserted-by":"crossref","first-page":"905","DOI":"10.1007\/s40306-018-00306-1","volume":"44","author":"Babaei S.","year":"2019","journal-title":"Acta Math. Vietnam."},{"key":"S1793830922501051BIB003","first-page":"381","volume":"46","author":"Bakhtyiari M.","year":"2017","journal-title":"Hokkaido Math. J."},{"key":"S1793830922501051BIB004","doi-asserted-by":"crossref","first-page":"208","DOI":"10.1016\/0021-8693(88)90202-5","volume":"116","author":"Beck I.","year":"1988","journal-title":"J. Algebra"},{"volume-title":"Graph Theory","year":"1997","author":"Diestel R.","key":"S1793830922501051BIB005"},{"key":"S1793830922501051BIB006","first-page":"203","volume":"23","author":"Lu C. P.","year":"1997","journal-title":"Houston J. Math."},{"key":"S1793830922501051BIB007","doi-asserted-by":"crossref","first-page":"39","DOI":"10.2298\/PIM1920039N","volume":"106","author":"Nozari K.","year":"2019","journal-title":"Publ. Inst. Math."},{"key":"S1793830922501051BIB008","doi-asserted-by":"crossref","first-page":"577","DOI":"10.7494\/OpMath.2019.39.4.577","volume":"4","author":"Pejman B.","year":"2019","journal-title":"Opuscula Math."},{"key":"S1793830922501051BIB009","volume-title":"Steps in Commutative Algebra","author":"Sharp R. Y.","year":"2000","edition":"2"}],"container-title":["Discrete Mathematics, Algorithms and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S1793830922501051","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,30]],"date-time":"2023-04-30T10:35:43Z","timestamp":1682850943000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/10.1142\/S1793830922501051"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,24]]},"references-count":9,"journal-issue":{"issue":"04","published-print":{"date-parts":[[2023,5]]}},"alternative-id":["10.1142\/S1793830922501051"],"URL":"https:\/\/doi.org\/10.1142\/s1793830922501051","relation":{},"ISSN":["1793-8309","1793-8317"],"issn-type":[{"type":"print","value":"1793-8309"},{"type":"electronic","value":"1793-8317"}],"subject":[],"published":{"date-parts":[[2022,5,24]]},"article-number":"2250105"}}