2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/122531We introduce the notion of bi-monoid in general monoidal category generalizing by this the notion of bialgebra. In the case of bimodules over a noncommutative algebra, we obtain a compatibility condition between ring and coring whenever both structures admit the same underlying bimodule.Rings and Algebras16W30; 16D20; 16D90text