Study of semi-compact spaces in commutative rings

In this paper, we investigate the structure of the prime spectrum Spec(R) of a commutative ring R, focusing on its behavior under the Semi-compact topology. We examine the topological characteristics of Spec(R) when endowed with the classical Zariski topology, where closed sets are of the form:v(I)={pϵ Sece(R):I⊆p for an ideal I⊆R } Which defines the closed sets of the Zariski topology on Spec(R)We introduce the notion of the primal Z-uncore to address separation issues inherent to Zariski spaces and explore conditions under which Spec(R) becomes Semi-compact, particularly for Z-ideal determined rings. Our results deepen the understanding of the relationship between topological compactness and algebraic structure in commutative algebra.