PICTURE FUZZY CHARACTERISTIC AND PICTURE FUZZY DIVISOR OF ZERO IN A RING ALONG WITH THEIR APPLICATIONS IN REAL LIFE SITUATIONS

The paper provides the concept of a picture fuzzy ideal over a ring and illustrates it with example. Additionally, it introduces the notions of picture fuzzy characteristic and picture fuzzy divisor of zero in a ring with respect to some picture fuzzy subring, investigating their related properties. The conditions under which picture fuzzy characteristic and picture fuzzy divisor of zero coincide with ordinary characteristic and ordinary divisor of zero are explicitly stated. It is demonstrated that the unit element in a ring, under some certain conditions, is not a picture fuzzy divisor of zero. Moreover, the picture fuzzy characteristic of the Cartesian product of two rings over the Cartesian product of two picture fuzzy subrings is calculated. Lastly, applications of PFCh and PFD of zero in Customers' feedback and Customers' sentiment analysis are presented \textbf{Significant note of the work:} Characteristic and divisor of zero are two foundational concepts in Abstract Algebra, playing a pivotal role in understanding the structure and behavior of rings. In this paper, these classical notions are extended and explored within the picture fuzzy environment, an advanced framework that builds upon fuzzy sets and intuitionistic fuzzy sets. By generalizing these concepts, the study bridges the gap between abstract algebraic theory and the flexible, nuanced reasoning enabled by picture fuzziness. Key properties of picture fuzzy characteristic and picture fuzzy divisor of zero are thoroughly investigated, revealing how these concepts retain their essence while adapting to the picture fuzzy context. Furthermore, the abstract notions of characteristic and divisor of zero are connected to real-world scenarios, demonstrating their relevance and applicability when interpreted through the lens of picture fuzziness. This work not only advances the theoretical understanding of algebraic structures but also opens new avenues for applying fuzzy algebra in solving practical problems, thereby inspiring innovative research in both mathematics and interdisciplinary fields.