Ahmed, Bayo MusaBakare, Gatta N.Usamot, Idayat F.2026-04-272026-04-272026-04-14Ahmed, B. M., Bakare, G. N. & Usamot, I. F. (2026). ON THE SEMIGROUP RING OF THE RHOTRIX BICYCLIC SEMIGROUP. Journal of Science, Technology, Mathematics and Education, 21(1), 236 - 245.https://uilspace.unilorin.edu.ng/handle/123456789/17717In this paper, we study the algebraic structure of the semigroup ring associated with the rhotrix bicyclic semigroup . Let be a ring and the rhotrix bicyclic semigroup defined on ordered pairs of sequences indexed by a fixed index set. We construct the semigroup ring and investigate some of its structural properties. In particular, we establish that is generally non-commutative and determine a family of idempotent elements arising from idempotents of . The ordering of these idempotents is described componentwise, and this ordering induces a natural hierarchy within the semigroup ring. Furthermore, several classes of subrings of are identified, including subrings generated by subsemigroups, idempotents, and corner subrings determined by idempotent elements. These results reveal how the structural properties of the rhotrix bicyclic semigroup influence the internal structure of its semigroup ring and provide a foundation for further study of ideals and related algebraic properties.enCorner subringsidempotent elementsordered idempotentsrhotrix bicyclic semigroupsemigroup ringsubrings.Article