Iranian Mathematical Society Journal of the Iranian Mathematical Society 2717-1612 5 2 2024 03 01 An overview of $z$-ideals and $z^\circ$-ideals 127 170 198518 10.30504/jims.2024.419987.1149 EN F. Azarpanah Department of Mathematics, Shahid Chamran University of Ahvaz Journal Article 2023 10 09 Overall‎, ‎$z$-ideals and $z^\circ$-ideals in $C(X)$‎, ‎the ring of all real-valued continuous functions on a space $X$‎, ‎play a crucial role in the ideal structure of the ring‎, ‎exhibiting connections with prime ideals and offering insights into the interplay between algebraic and topological properties of the space $X$‎. ‎By exploring their characteristics and relationships with prime ideals‎, ‎we can better understand the intricate nature of these ideals and their impact on the structure of $C(X)$‎. ‎Studying $z$-ideals and $z^\circ$-ideals in reduced rings‎, ‎particularly in $C(X)$‎, ‎sheds light on the fundamental aspects of ring theory and topology‎, ‎highlighting the intricate connections between these two fields‎. ‎A pseudoprime $z$-ideal is prime‎, ‎and a prime ideal minimal over a $z$-ideal ($z^\circ$-ideal), ‎is also a $z$-ideal ($z^\circ$-ideal)‎. ‎Additionally‎, ‎the sum of a prime ideal and a $z$-ideal is a prime $z$-ideal‎, ‎and every $z$-ideal ($z^\circ$-ideal) is an intersection of prime $z$-ideals ($z^\circ$-ideals)‎. ‎Furthermore‎, ‎every ideal contains the largest $z$-ideal and is included in the smallest $z$-ideal‎. ‎These properties demonstrate the significance of $z$-ideals and $z^\circ$-ideals in the ideal structure of the ring $C(X)$ and their role in connecting the algebraic and topological properties of the space $X$‎. ‎By exploring these properties in reduced rings‎, ‎especially in $C(X)$‎, ‎we can appreciate the intricate relationship between the algebraic aspects of $C(X)$ and the topological characteristics of $X$‎. ‎The elegance and effectiveness of $z$-ideals and $z^\circ$-ideals in this context highlight their importance in understanding the intersection of algebra and topology within $C(X)$‎. ‎The study of $z$-ideals and $z^\circ$-ideals in reduced rings‎, ‎particularly in $C(X)$‎, ‎stands out for its elegance and effectiveness in elucidating the ideal structure of the ring $C(X)$‎. ‎Inasmuch as $z$-ideals and $z^\circ$-ideals are both algebraic and topological objects‎, ‎they play a crucial role in bridging the gap between the algebraic properties of $C(X)$ and the topological properties of the space $X$‎. ‎This article aims to compile and explore the properties of $z$-ideals and $z^\circ$-ideals in $C(X)$‎, ‎emphasizing their significance in understanding the connections between algebraic and topological aspects within this framework‎. https://jims.ims.ir/article_198518_7e7713a79b0a18f474bf560c3fdc51f1.pdf