5f055225-10fb-4821-99b0-764b2c2dfb3e20250616051049068wseas:wseasmdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON MATHEMATICS2224-28801109-276910.37394/23206http://wseas.org/wseas/cms.action?id=4051120202512020252410.37394/23206.2025.24https://wseas.com/journals/mathematics/2025.phpJovannie R.CastilloDepartment of Mathematics and Natural Sciences North Eastern Mindanao State University 8300 Tandag City PHILIPPINESIn this paper, we introduce the concept of a ring-like algebraic structure associated with a B−algebra, termed a BA−ringoid, and establish several of its fundamental properties that are reminiscent of classical rings. In addition, we construct quotient structures for BA−ringoids by defining suitable equivalence relations, offering a broader perspective on how these systems can be partitioned and analyzed.6162025616202541642339https://wseas.com/journals/mathematics/2025/a785106-2200.pdf10.37394/23206.2025.24.39https://wseas.com/journals/mathematics/2025/a785106-2200.pdfBruck R.H. A survey of Binary Systems. Springer, New York, 1971. https: //doi.org/10.1007/978-3-662-43119-1 Neggers J., and Kim H.S. On d-algebras, Math. Slovaca vol. 49 (1999). http://dml.cz/dmlcz/129981 10.2478/s12175-007-0001-zAllen P.J, Kim H., and Neggers J., Companion d-algebras, Math. Slovaca 57 (2007), no. 2, 93-106. https: //doi.org/10.2478/s12175-007-0001-z Naingue L.D. and Vilela J.P., On companion BAlgebras. European Journal of Pure and Applied Mathematics. Vol.12, No. 3 (2019). https://DOI: 10.2478/s12175-007-0001-z Neggers J. and Kim H.S., On B-algebras . Matematiki Vesnik, 54. pp21-29, (2002). https://www.researchgate.net/ publication/267164912_On_B-algebras Kondo M. and Jun Y., The class of B-algebras coincides with the class of groups. Scientiae Mathematicae Japonicae Online, 2:175–177, 2002. https://www.jams.or.jp/scm/contents/ Vol-7-2/7-20.pdf Allen P.J., Neggers J., and Kim H.S., B− Algebras and Groups. Scientiae Mathematicae Japonicae. Vol.9 (2023) https://www.jams.or.jp/scm/contents/ Vol-9-2/9-17.pdf Cho J. Kim H.S. On B-algebras and Quasigroups. Quasigroups and Related Systems, vol.7(2001). https://www.math.md/files/qrs/v8-n1/ v8-n1-(pp1-6).pd 10.12988/ams.2017.75159Abdullah H. and Atshan A., Complete ideal and n-ideal of B-algebra. Applied Mathematical Sciences, Vol 11 No. 35 ( 2017). https: //doi.org/10.12988/ams.2017.75159 10.1515/dema-2008-0203Kim C. and Kim H., Another Axiomatization of B-algebras, Demonstratio Math. Vol. 41 No. (2008). 259-262. https: //doi.org/10.1515/dema-2008-0203 Neggers J. and Kim H.S., A Fundamental Theorem of B- homomorphism for B-algebras International Mathematics Journal 2. pp 207-214, (2002). https://cir.nii.ac.jp/crid/ 1570572700535790336?lang=en Alsheri N.O., Derivations of B- Algebras. JKAU Scie. Vol 22, N0. 1 (2010). DOI:10.4197/Sci.22-1.5 Kim H.S. and Park H.H, On 0− Commutative B-Algebras. Scientiae Mathematicae Japonicae Online, e-2005, 31–36. ams.jp/scm/contents/e-2005-1/2005-4. pdf 10.52547/hatef.jahla.3.4.3Neggers J, Kim H.S., and Rezaei A. On Ringoids. Journal of Algebraic Hyperstructures and Logical Algebras vol. 3, No.4 (2022). https: //jahla.hatef.ac.ir/article_160777_ 62f8d515e82e60f645dbb2ec3bf2be44.pdf