Approximately Prime Rings and Prime Ideals
Maram Almahariq, James Francis Peters, Tane Vergili
TL;DR
This work extends classical prime ring and prime ideal theory to descriptive proximity settings by introducing approximately prime rings and approximately ideals within proximal relator spaces. It defines the product of approximately ideals and the approximately direct product of rings, and proves foundational properties such as the preservation of the approximately ideal structure under product and key prime-related divisibility results. The authors also characterize approximately prime rings via quotients $R/I$ being approximately integral domains and discuss approximately irreducible elements and approximately principal primes to connect primality with invertibility concepts. The results provide a framework for rough, proximity-based algebraic reasoning, with potential applications to digital imagery and descriptive proximity analysis.
Abstract
This article focuses on approximately prime rings and approximately prime ideals in proximal relator spaces, especially in descriptive proximity spaces. In particular, we define some binary operations, including the product of two approximately prime ideals and the direct product of approximately prime rings, and study the approximately principle prime ideals. Moreover, we introduce some fundamental properties of these approximately algebraic structures.