{"id":"https://openalex.org/W4407965728","doi":"https://doi.org/10.3390/axioms14030170","title":"Exploring the Embedding of the Extended Zero-Divisor Graph of Commutative Rings","display_name":"Exploring the Embedding of the Extended Zero-Divisor Graph of Commutative Rings","publication_year":2025,"publication_date":"2025-02-26","ids":{"openalex":"https://openalex.org/W4407965728","doi":"https://doi.org/10.3390/axioms14030170"},"language":"en","primary_location":{"id":"doi:10.3390/axioms14030170","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms14030170","pdf_url":"https://www.mdpi.com/2075-1680/14/3/170/pdf?version=1740576153","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2075-1680/14/3/170/pdf?version=1740576153","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5003744154","display_name":"Ali Al Khabyah","orcid":"https://orcid.org/0000-0002-4167-3119"},"institutions":[{"id":"https://openalex.org/I82952536","display_name":"King Khalid University","ror":"https://ror.org/052kwzs30","country_code":"SA","type":"education","lineage":["https://openalex.org/I82952536"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Ali Al Khabyah","raw_affiliation_strings":["Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia","institution_ids":["https://openalex.org/I82952536"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5032022086","display_name":"Moin A. Ansari","orcid":"https://orcid.org/0000-0002-1175-9704"},"institutions":[{"id":"https://openalex.org/I91450128","display_name":"Jazan University","ror":"https://ror.org/02bjnq803","country_code":"SA","type":"education","lineage":["https://openalex.org/I91450128"]}],"countries":["SA"],"is_corresponding":true,"raw_author_name":"Moin A. Ansari","raw_affiliation_strings":["Department of Mathematics, College of Science, Jazan University, P.O. Box 114, Jazan 45142, Saudi Arabia"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, College of Science, Jazan University, P.O. Box 114, Jazan 45142, Saudi Arabia","institution_ids":["https://openalex.org/I91450128"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5032022086"],"corresponding_institution_ids":["https://openalex.org/I91450128"],"apc_list":{"value":1400,"currency":"CHF","value_usd":1515},"apc_paid":{"value":1400,"currency":"CHF","value_usd":1515},"fwci":5.9683,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.91637182,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":91,"max":95},"biblio":{"volume":"14","issue":"3","first_page":"170","last_page":"170"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11703","display_name":"Rings, Modules, and Algebras","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11703","display_name":"Rings, Modules, and Algebras","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11673","display_name":"Advanced Topics in Algebra","score":0.9959999918937683,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10287","display_name":"Algebraic structures and combinatorial models","score":0.989300012588501,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/zero-divisor","display_name":"Zero divisor","score":0.9285777807235718},{"id":"https://openalex.org/keywords/commutative-ring","display_name":"Commutative ring","score":0.6542288064956665},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6107895374298096},{"id":"https://openalex.org/keywords/embedding","display_name":"Embedding","score":0.5876916646957397},{"id":"https://openalex.org/keywords/zero","display_name":"Zero (linguistics)","score":0.5194994211196899},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.46055495738983154},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.45772087574005127},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.31285879015922546},{"id":"https://openalex.org/keywords/commutative-property","display_name":"Commutative property","score":0.31117066740989685},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.17027395963668823},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.07916897535324097},{"id":"https://openalex.org/keywords/philosophy","display_name":"Philosophy","score":0.07290798425674438},{"id":"https://openalex.org/keywords/linguistics","display_name":"Linguistics","score":0.06723770499229431}],"concepts":[{"id":"https://openalex.org/C9555016","wikidata":"https://www.wikidata.org/wiki/Q828111","display_name":"Zero divisor","level":2,"score":0.9285777807235718},{"id":"https://openalex.org/C161491579","wikidata":"https://www.wikidata.org/wiki/Q858656","display_name":"Commutative ring","level":3,"score":0.6542288064956665},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6107895374298096},{"id":"https://openalex.org/C41608201","wikidata":"https://www.wikidata.org/wiki/Q980509","display_name":"Embedding","level":2,"score":0.5876916646957397},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.5194994211196899},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.46055495738983154},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.45772087574005127},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.31285879015922546},{"id":"https://openalex.org/C183778304","wikidata":"https://www.wikidata.org/wiki/Q165474","display_name":"Commutative property","level":2,"score":0.31117066740989685},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.17027395963668823},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.07916897535324097},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.07290798425674438},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.06723770499229431}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/axioms14030170","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms14030170","pdf_url":"https://www.mdpi.com/2075-1680/14/3/170/pdf?version=1740576153","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:30ed354b4a2644f49e80ae96185b0635","is_oa":true,"landing_page_url":"https://doaj.org/article/30ed354b4a2644f49e80ae96185b0635","pdf_url":null,"source":{"id":"https://openalex.org/S112646816","display_name":"SHILAP Revista de lepidopterolog\u00eda","issn_l":"0300-5267","issn":["0300-5267","2340-4078"],"is_oa":true,"is_in_doaj":true,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Axioms, Vol 14, Iss 3, p 170 (2025)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/axioms14030170","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms14030170","pdf_url":"https://www.mdpi.com/2075-1680/14/3/170/pdf?version=1740576153","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1153394551","display_name":null,"funder_award_id":"RGP2/339/45","funder_id":"https://openalex.org/F4320324433","funder_display_name":"King Khalid University"}],"funders":[{"id":"https://openalex.org/F4320324433","display_name":"King Khalid University","ror":"https://ror.org/052kwzs30"}],"has_content":{"pdf":true,"grobid_xml":false},"content_urls":{"pdf":"https://content.openalex.org/works/W4407965728.pdf"},"referenced_works_count":25,"referenced_works":["https://openalex.org/W1577244476","https://openalex.org/W1975461789","https://openalex.org/W1987023204","https://openalex.org/W1998294201","https://openalex.org/W2002257546","https://openalex.org/W2047344301","https://openalex.org/W2141559405","https://openalex.org/W2169604893","https://openalex.org/W2587107303","https://openalex.org/W2767305887","https://openalex.org/W2938481609","https://openalex.org/W2959544132","https://openalex.org/W2973509842","https://openalex.org/W3169746652","https://openalex.org/W3209391632","https://openalex.org/W3213265554","https://openalex.org/W4205993708","https://openalex.org/W4224300912","https://openalex.org/W4225982668","https://openalex.org/W4310255316","https://openalex.org/W4401213539","https://openalex.org/W4403750111","https://openalex.org/W6745628868","https://openalex.org/W6806559982","https://openalex.org/W6983412328"],"related_works":["https://openalex.org/W2147421839","https://openalex.org/W2963575964","https://openalex.org/W2162054958","https://openalex.org/W2372618431","https://openalex.org/W2385206495","https://openalex.org/W2097947972","https://openalex.org/W2120359860","https://openalex.org/W4397030742","https://openalex.org/W3113897730","https://openalex.org/W3209879031"],"abstract_inverted_index":{"Rc":[0,15,83,123],"represents":[1],"commutative":[2],"rings":[3],"that":[4],"have":[5],"unity":[6],"elements.":[7],"The":[8,41,71],"collection":[9],"of":[10,30,37,50,91,112,134,142],"all":[11],"zero-divisor":[12,24,48,94,105,127],"elements":[13,49],"in":[14,39,61],"are":[16,59],"represented":[17],"by":[18,26],"D(Rc).":[19],"We":[20],"denote":[21],"an":[22,131],"extended":[23,93,104,126],"graph":[25,33,42,106,128],"the":[27,47,81,88,103,125,139],"notation":[28],"\u2138\u2032(Rc)":[29,43,62,129,143],"Rc.":[31,51],"This":[32,96],"has":[34,107,130],"a":[35,108],"set":[36],"vertices":[38,55],"D(Rc)*.":[40],"illustrates":[44],"interactions":[45],"among":[46],"Specifically,":[52],"two":[53],"different":[54],"u":[56],"and":[57],"y":[58],"connected":[60],"iff":[63],"uRc\u2229Ann(y)":[64],"is":[65,69,77,85],"non-null":[66],"or":[67,110],"yRc\u2229Ann(u)":[68],"non-null.":[70],"main":[72],"idea":[73],"for":[74,87,144],"this":[75,115],"work":[76,116],"to":[78,118],"systematically":[79],"analyze":[80],"ring":[82,122],"which":[84],"finite":[86,121],"unique":[89],"aspect":[90],"their":[92],"graph.":[95],"study":[97],"particularly":[98],"focuses":[99],"on":[100],"instances":[101],"where":[102],"genus":[109,145],"crosscap":[111],"two.":[113,135],"Furthermore,":[114],"aims":[117],"thoroughly":[119],"characterize":[120],"wherein":[124],"outerplanarity":[132],"index":[133],"Finally,":[136],"we":[137],"determine":[138],"book":[140],"thickness":[141],"at":[146],"most":[147],"one.":[148]},"counts_by_year":[{"year":2025,"cited_by_count":1}],"updated_date":"2025-12-21T01:58:51.020947","created_date":"2025-10-10T00:00:00"}
