{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,21]],"date-time":"2025-11-21T12:27:11Z","timestamp":1763728031653,"version":"build-2065373602"},"reference-count":52,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,8,1]],"date-time":"2021-08-01T00:00:00Z","timestamp":1627776000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The role of symmetry in ring theory is universally recognized. The most directly definable universal relation in a symmetric set theory is isomorphism. This article develops a certain structure of bipolar fuzzy subrings, including bipolar fuzzy quotient ring, bipolar fuzzy ring homomorphism, and bipolar fuzzy ring isomorphism. We define (\u03b1,\u03b2)-cut of bipolar fuzzy set and investigate the algebraic attributions of this phenomenon. We also define the support set of bipolar fuzzy set and prove various important properties relating to this concept. Additionally, we define bipolar fuzzy homomorphism by using the notion of natural ring homomorphism. We also establish a bipolar fuzzy homomorphism between bipolar fuzzy subring of the quotient ring and bipolar fuzzy subring of this ring. We constituted a significant relationship between two bipolar fuzzy subrings of quotient rings under a given bipolar fuzzy surjective homomorphism. We present the construction of an induced bipolar fuzzy isomorphism between two related bipolar fuzzy subrings. Moreover, to discuss the symmetry between two bipolar fuzzy subrings, we present three fundamental theorems of bipolar fuzzy isomorphism.<\/jats:p>","DOI":"10.3390\/sym13081397","type":"journal-article","created":{"date-parts":[[2021,8,1]],"date-time":"2021-08-01T21:46:44Z","timestamp":1627854404000},"page":"1397","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["A Certain Structure of Bipolar Fuzzy Subrings"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7845-7751","authenticated-orcid":false,"given":"Hanan","family":"Alolaiyan","sequence":"first","affiliation":[{"name":"Department of Mathematics, King Saud University, Riyadh 11451, Saudi Arabia"}]},{"given":"Muhammad Haris","family":"Mateen","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of the Punjab, Lahore 54590, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8522-1942","authenticated-orcid":false,"given":"Dragan","family":"Pamucar","sequence":"additional","affiliation":[{"name":"Department of Logistics, Military Academy, University of Defence in Belgarde, 11000 Belgarde, Serbia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1071-2808","authenticated-orcid":false,"given":"Muhammad Khalid","family":"Mahmmod","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of the Punjab, Lahore 54590, Pakistan"}]},{"given":"Farrukh","family":"Arslan","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, University of Engineering and Technology, Lahore 54890, Pakistan"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Inf. 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