Some topological considerations on semistar operations

Abstract

We consider properties and applications of a new topology, called the Zariski topology, on the space $\mathrm{SStar}(A)$ of all the semistar operations on an integral domain $A$. We prove that the set of all overrings of $A$, endowed with the classical Zariski topology, is homeomorphic to a subspace of $\mathrm{SStar}(A)$. The topology on $\mathrm{SStar}(A)$ provides a general theory, through which we see several algebraic properties of semistar operation as very particular cases of our construction. Moreover, we show that the subspace ${\mathrm{SStar}}_f(A)$ of all the semistar operations of finite type on $A$ is a spectral space.

17 pages. To appear in J. Algebra

References in corpus (1)

• Structure on the set of closure operations of a commutative ring

Cited by in corpus (6)

• The space of homogeneous preserving semistar operations on graded domains
• Isolated points of the Zariski space
• Sharpness and semistar operations in Pruefer-like domains
• Topological properties of semigroup primes of a commutative ring
• Suprema in spectral spaces and the constructible closure
• Intersections of quotient rings and Pruefer v-multiplication domains