• 대한수학회보
• /
• 제61권1호
• /
• pp.83-92
• /
• 2024

Let R be a commutative ring with nonzero identity and M be an R-module. In this paper, we first introduce the concept of S-idempotent element of R. Then we give a relation between S-idempotents of R and clopen sets of S-Zariski topology. After that we define S-pure ideal which is a generalization of the notion of pure ideal. In fact, every pure ideal is S-pure but the converse may not be true. Afterwards, we show that there is a relation between S-pure ideals of R and closed sets of S-Zariski topology that are stable under generalization.

• 호남수학학술지
• /
• 제28권3호
• /
• pp.365-376
• /
• 2006

Let R be a commutative ring and let M be an R-module. Let X = $Spec_R(M)$ be the prime spectrum of M with Zariski topology. In this paper, by using the topological properties of X, we will obtain some conditions under which $Max_R(M)=Spec_R(M)$.

• Korean Journal of Mathematics
• /
• 제28권3호
• /
• pp.439-447
• /
• 2020

Let M be a lattice module over a C-lattice L. Let Spec^s(M) be the collection of all second elements of M. In this paper, we consider a topology on Spec^s(M), called the second classical Zariski topology as a generalization of concepts in modules and investigate the interplay between the algebraic properties of a lattice module M and the topological properties of Spec^s(M). We investigate this topological space from the point of view of spectral spaces. We show that Spec^s(M) is always T₀-space and each finite irreducible closed subset of Spec^s(M) has a generic point.