A fundamental theorem of powerful set-valued for F-rough ring

TL;DR

This paper develops a new isomorphism theorem for F-rough rings using powerful set-valued homomorphisms, extending classical ring theory with set-valued approximation concepts.

Contribution

It introduces the first isomorphism theorem for F-rough rings, generalizing set-valued mappings in ring theory and analyzing properties of approximations.

Findings

Kernel of homomorphism is a subring.

Established properties of upper and lower approximations.

Proved the isomorphism theorem for F-rough rings.

Abstract

In this paper, we introduce the upper and lower approximations on the invers set-valued mapping and the approximations an established on a powerful set valued homomorphism from a ring R1 to power sets of a ring R2. Moreover, the properties of lower and upper approximations of a powerful set valued are studies. In addition, we will give a proof of the theorem of isomorphism over approximations F-rough ring as new result. However, we will prove the kernel of the powerful set-valued homomorphism is a subring of R1. Our result is introduce the first isomorphism theorem of ring as generalized the concept of the set valued mappings.