Abstract
This paper uses some logical algebraic elements to extend any ring into a non-commutative ring containing the original ring with many generalized substructures and special elements. On the other hand, we study the substructures of non-commutative logical rings such as AH-homomorphisms and AH-ideals with many examples that explain their algebraic validity. Also, we discuss the possibility of solving a linear Diophantine equation with two variables in the non-commutative logical ring of integers, where we present an easy algorithm to solve this kind of generalized Diophantine equation.
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