On the extensions of Goursat’s theorem to direct products of n R-algebras

In algebraic theory, there are groups, rings, modules over a ring, and algebra over rings. As a result, people try to develop theories that apply to groups to apply to other concepts. For example, Goursat’s theorem in groups related to the direct product of the two groups, then developed in the ring, the module over a ring, the algebra over a ring related to the direct product of the two rings, two modules over a ring, and two algebras over a ring, respectively. Can we extend Goursat’s theorem in algebras over a ring R (R-algebras) with respect to the direct product of n R-algebras? So, this paper will extend Goursat’s theorem to direct products of n R-algebras, exploring related properties and providing formal proof. The main result of this study demonstrates that every subalgebra in a direct product of n R-algebras can be uniquely determined by n − 1 R-algebra epimorphisms of an algebra to a factor algebra.