{"id":"https://openalex.org/W7215219249","doi":null,"title":"Some Commutativity Theorems for Non-Associative Rings","display_name":"Some Commutativity Theorems for Non-Associative Rings","publication_year":2017,"publication_date":"2017-06-15","ids":{"openalex":"https://openalex.org/W7215219249"},"language":"en","primary_location":{"id":"pmh:oai:ojs2.ijmaa.in:article/831","doi":null,"is_oa":true,"landing_page_url":"https://ijmaa.in/index.php/ijmaa/article/view/831","pdf_url":"https://ijmaa.in/index.php/ijmaa/article/download/831/820","source":{"id":"https://openalex.org/S7407065868","display_name":"International Journal of Mathematics And its Applications","issn_l":"2347-1557","issn":["2347-1557"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by-nc","license_id":"https://openalex.org/licenses/cc-by-nc","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"2347-1557","raw_type":"info:eu-repo/semantics/article"},"volume":null,"type":"article","is_traditional":true,"indexed_in":[],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://ijmaa.in/index.php/ijmaa/article/download/831/820","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5053078098","display_name":"Y.Madana Mohana Reddy","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I119068399","display_name":"Rayalaseema University","ror":"https://ror.org/00nphdr35","country_code":"IN","type":"education","lineage":["https://openalex.org/I119068399"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Y. Madana Mohana Reddy","raw_affiliation_strings":["Department of Mathematics, Rayalaseema University, Kurnool, Andhra Pradesh, India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Rayalaseema University, Kurnool, Andhra Pradesh, India","institution_ids":["https://openalex.org/I119068399"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5019057441","display_name":"G. Shobhalatha","orcid":"https://orcid.org/0000-0003-4964-7502","observed_orcids":["https://orcid.org/0000-0003-4964-7502"]},"institutions":[{"id":"https://openalex.org/I4031970","display_name":"Sri Krishnadevaraya University","ror":"https://ror.org/02fyxjb45","country_code":"IN","type":"education","lineage":["https://openalex.org/I4031970"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"G. Shobhalatha","raw_affiliation_strings":["Department of Mathematics, S.K.University, Anantapur, Andhra Pradesh, India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, S.K.University, Anantapur, Andhra Pradesh, India","institution_ids":["https://openalex.org/I4031970"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5154007381","display_name":"D. T. Hanumantha Reddy","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"D. V. Rami Reddy","raw_affiliation_strings":["Department of Mathematics, AVR & SVR Engineering College, Nandayal, Andhra Pradesh, India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, AVR & SVR Engineering College, Nandayal, Andhra Pradesh, India","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.0,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11703","display_name":"Rings, Modules, and Algebras","score":0.9984999895095825,"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":0.9984999895095825,"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":0.000699999975040555,"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/T12504","display_name":"Mathematics and Applications","score":0.0,"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/commutativity-theorems","display_name":"commutativity theorems","score":0.8920000195503235},{"id":"https://openalex.org/keywords/nonassociative-rings","display_name":"nonassociative rings","score":0.9980000257492065},{"id":"https://openalex.org/keywords/commutativity-of-rings","display_name":"commutativity of rings","score":0.4749999940395355},{"id":"https://openalex.org/keywords/associative-rings","display_name":"associative rings","score":0.5649999976158142},{"id":"https://openalex.org/keywords/2-torsion-free-rings","display_name":"2-torsion free rings","score":0.8640000224113464},{"id":"https://openalex.org/keywords/center-of-a-ring","display_name":"center of a ring","score":0.7730000019073486},{"id":"https://openalex.org/keywords/lie-brackets","display_name":"Lie brackets","score":0.4050000011920929}],"concepts":[{"id":"https://openalex.org/C2780378348","wikidata":"https://www.wikidata.org/wiki/Q25351438","display_name":"Ring (chemistry)","level":2,"score":0.7294999957084656},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7107999920845032},{"id":"https://openalex.org/C100044566","wikidata":"https://www.wikidata.org/wiki/Q6423024","display_name":"Reduced ring","level":5,"score":0.6891999840736389},{"id":"https://openalex.org/C183778304","wikidata":"https://www.wikidata.org/wiki/Q165474","display_name":"Commutative property","level":2,"score":0.6769000291824341},{"id":"https://openalex.org/C125225535","wikidata":"https://www.wikidata.org/wiki/Q7049224","display_name":"Noncommutative ring","level":3,"score":0.609499990940094},{"id":"https://openalex.org/C156923205","wikidata":"https://www.wikidata.org/wiki/Q2905043","display_name":"Principal ideal ring","level":4,"score":0.5981000065803528},{"id":"https://openalex.org/C161491579","wikidata":"https://www.wikidata.org/wiki/Q858656","display_name":"Commutative ring","level":3,"score":0.5745000243186951},{"id":"https://openalex.org/C2780129039","wikidata":"https://www.wikidata.org/wiki/Q1931107","display_name":"Section (typography)","level":2,"score":0.5331000089645386},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.44369998574256897},{"id":"https://openalex.org/C170320730","wikidata":"https://www.wikidata.org/wiki/Q1142699","display_name":"Primary ideal","level":5,"score":0.39739999175071716},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.35350000858306885},{"id":"https://openalex.org/C21308566","wikidata":"https://www.wikidata.org/wiki/Q7169365","display_name":"Permutation (music)","level":2,"score":0.3479999899864197},{"id":"https://openalex.org/C202836834","wikidata":"https://www.wikidata.org/wiki/Q2895708","display_name":"Primitive ring","level":5,"score":0.34779998660087585},{"id":"https://openalex.org/C83920498","wikidata":"https://www.wikidata.org/wiki/Q2326800","display_name":"Simple ring","level":5,"score":0.34610000252723694},{"id":"https://openalex.org/C159423971","wikidata":"https://www.wikidata.org/wiki/Q177251","display_name":"Associative property","level":2,"score":0.3271999955177307},{"id":"https://openalex.org/C128134353","wikidata":"https://www.wikidata.org/wiki/Q2634401","display_name":"Boolean ring","level":5,"score":0.3203999996185303},{"id":"https://openalex.org/C2779463800","wikidata":"https://www.wikidata.org/wiki/Q5062222","display_name":"Center (category theory)","level":2,"score":0.3172999918460846},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3084999918937683},{"id":"https://openalex.org/C190552800","wikidata":"https://www.wikidata.org/wiki/Q2895315","display_name":"Von Neumann regular ring","level":3,"score":0.2897000014781952}],"mesh":[],"locations_count":1,"locations":[{"id":"pmh:oai:ojs2.ijmaa.in:article/831","doi":null,"is_oa":true,"landing_page_url":"https://ijmaa.in/index.php/ijmaa/article/view/831","pdf_url":"https://ijmaa.in/index.php/ijmaa/article/download/831/820","source":{"id":"https://openalex.org/S7407065868","display_name":"International Journal of Mathematics And its Applications","issn_l":"2347-1557","issn":["2347-1557"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by-nc","license_id":"https://openalex.org/licenses/cc-by-nc","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"2347-1557","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"pmh:oai:ojs2.ijmaa.in:article/831","doi":null,"is_oa":true,"landing_page_url":"https://ijmaa.in/index.php/ijmaa/article/view/831","pdf_url":"https://ijmaa.in/index.php/ijmaa/article/download/831/820","source":{"id":"https://openalex.org/S7407065868","display_name":"International Journal of Mathematics And its Applications","issn_l":"2347-1557","issn":["2347-1557"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by-nc","license_id":"https://openalex.org/licenses/cc-by-nc","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"2347-1557","raw_type":"info:eu-repo/semantics/article"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7215219249.pdf","grobid_xml":"https://content.openalex.org/works/W7215219249.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2228738655","https://openalex.org/W3009890612","https://openalex.org/W2771455137","https://openalex.org/W7215156618","https://openalex.org/W2116694668","https://openalex.org/W2380991363","https://openalex.org/W3159210059","https://openalex.org/W7096757151","https://openalex.org/W2123470368","https://openalex.org/W2116049525"],"abstract_inverted_index":{"In":[0,13],"this":[1,50,166],"section":[2,167],"we":[3,77,102,168],"prove":[4,78,103],"some":[5,171,187],"elementary":[6,15],"results":[7],"on":[8],"commutativity":[9,16,105],"of":[10,98,106,116,165,178,214],"non-associative":[11,28,110],"rings.":[12],"an":[14,192],"theorem":[17],"for":[18,38,155],"rings,":[19],"Johnsen,":[20],"Outcalt":[21],"and":[22,64,74,90,181],"Yaqub":[23],"[1]":[24],"showed":[25],"that":[26,53,69,79,175,191],"a":[27,55,107,197],"ring":[29,56,71,82,100,111,194,199],"R":[30,43,112,195],"with":[31],"unity":[32,180],"satisfying":[33,113],"$\\left(xy\\right)^{2}":[34,85,121,125],"=x^{2}":[35,86],"y^{2}":[36,87,139,143],"$":[37],"all":[39,156],"x,":[40,157,215],"y":[41],"in":[42,83,160,186,200,218],"is":[44,57,62,72,196,211],"necessarily":[45],"commutative.":[46],"Boers":[47],"[2]":[48],"extended":[49],"to":[51],"show":[52,174],"such":[54],"also":[58,169],"associative":[59],"provided":[60],"it":[61],"2":[63,73],"3-torsion":[65,75],"free.":[66],"Without":[67],"using":[68],"the":[70,96,99,104,117,163,176,179],"free,":[76],"any":[80,114,212],"assosymmetric":[81,193],"which":[84,173,201],"$is":[88],"commutative":[89],"associative.":[91],"Further,":[92],"if":[93],"Z(R)":[94],"denotes":[95],"center":[97],"R,":[101],"2-torsion":[108,182],"free":[109,183],"one":[115],"following":[118],"identities:\\\\":[119],"(1).":[120],"\\in":[122],"Z\\left(R\\right)$\\\\":[123,127,131],"(2).":[124],"-xy\\in":[126],"(3).":[128],"$\\left(\\left(xy\\right)z\\right)^{2}":[129],"-\\left(xy\\right)z\\in":[130],"(4).":[132],"$\\left[\\left(xy\\right)^{2}":[133,135,146,151],"-yx,x\\right]=0$or":[134],"-yx,y\\right]=0$\\\\":[136],"(5).":[137],"$\\left[x^{2}":[138,142],"-xy,x\\right]=0$":[140],"or":[141,150],"-xy,y\\right]=0$\\\\":[144],"(6).":[145],"-x^{2}":[147,152],"y-xy^{2}":[148,153],"+xy,x\\right]=0$":[149],"+xy,y\\right]=0$":[154],"y,":[158,203,216],"z":[159,217],"R.\\\\":[161],"At":[162],"end":[164],"give":[170],"example":[172],"existence":[177],"are":[184],"essential":[185],"results.":[188],"We":[189],"know":[190],"non-associtative":[198],"$(x,":[202],"z)":[204],"=":[205],"(P(x),":[206],"P(y),":[207],"P(z))$,":[208],"where":[209],"P":[210],"permutation":[213],"R.":[219]},"counts_by_year":[],"updated_date":"2026-10-05T09:50:58.205359","created_date":"2026-10-02T00:00:00"}