Polynomial Extensions of Fuzzy Baer and Fuzzy Quasi-Baer Subrings - Kute, Dinesh; Warke, Arundhati; Kharinar, Anil (Communications in Mathematics and Applications, 2025)
This paper introduces and investigates the notion of fuzzy quasi-Baer subrings, establishing fundamental properties relative to classical fuzzy algebraic structures. We demonstrate a hierarchical relationship between fuzzy Baer and fuzzy quasi-Baer subrings, proving that while every fuzzy Baer subring is quasi-Baer, the converse fails in general. The study focuses on polynomial extensions of these structures, demonstrating that the quasi-Baer property is preserved under polynomial extensions, unlike the Baer property. A key result establishes that when a polynomial extension of a fuzzy subring is quasi-Baer, the original fuzzy subring must necessarily be Baer. These findings reveal important structural constraints in fuzzy ring theory and provide new insights into the behavior of annihilator conditions in fuzzy algebraic systems. The work extends classical Baer and quasi-Baer ring theory to the fuzzy setting, while highlighting crucial distinctions between these concepts in polynomial extensions. Our results contribute to the growing body of knowledge in fuzzy algebra and open new directions for research in fuzzy homological algebra and categorical generalizations of noncommutative ring-theoretic properties.
Keywords: Fuzzy subring; Fuzzy Baer subring; Fuzzy quasi-Baer subring