In this paper, we study a generalization of z-IDEALS in the ring C (X) of continuous real valued functions on a completely regular Hausdorff space X. The notion of a weak ideal and naturally a weak z-ideal and a prime weak ideal are introduced and it turns out that they behave such as z -IDEALS in C (X).

In this paper, we introduce the notion of hybrid bi-IDEALS in semigroups and investigate some of their important properties. We also give various equivalent conditions for a semigroup to be regular and hybrid structures to be hybrid bi-IDEALS of S.

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.

It is well known that the sum of two z-IDEALS in C(X) is either C(X) or a z-ideal. The main aim of this paper is to study the sum of strongly z-IDEALS in RL, the ring of real-valued continuous functions on a frame L. For every ideal I in RL, we introduce the biggest strongly z-ideal included in I and the smallest strongly z-ideal containing I, denoted by Isz and Isz, respectively. We study some properties of Isz and Isz: Also, it is observed that the sum of any family of minimal prime IDEALS in the ring RL is either RL or a prime strongly z-ideal in RL. In particular, we show that the sum of two prime IDEALS in RL which are not chains is a prime strongly z-ideal.

In this paper, we introduce the concepts of J-prime IDEALS and MJ-IDEALS in posets, and obtain some of their interesting characterizations in posets. Furthermore, we discuss the properties of J-IDEALS that are analogous to J-prime IDEALS and MJ-IDEALS in posets. Finally, we establish a set of equivalent conditions for an ideal in a poset P containing an ideal J is an J-ideal, and for a semi-prime ideal J to be an MJ-ideal of P.