# Extension des operations semistar

## Extension of Semistar Operations

Gmiza Wafa
University of Monastir
Tunisia

Hizem Sana
University of Monastir
Tunisia

Publié le 7 mars 2023   DOI : 10.21494/ISTE.OP.2023.0934

### Keywords

Let RT be an extension of integral domains and ∗ be a semistar operation stable of finite type on R. We define a semistar operation ∗1 on T in the following way : for each nonzero T-submodule E of the quotient field K1 of T, let E∗1 = ∪ {E :K1 JT | J ∈ $\mathcal{F}$∗}, where K1 denotes the quotient field of T and $\mathcal{F}$∗ the localizing system associated to ∗. In this paper we investigate the basic properties of ∗1. Moreover, we show that the map $\varphi$ which associates to a semistar operation ∗ stable and of finite type on R, the semistar operation ∗1 is continuous. Furthermore, we give sufficient conditions for $\varphi$ to be a homeomorphism.

Let RT be an extension of integral domains and ∗ be a semistar operation stable of finite type on R. We define a semistar operation ∗1 on T in the following way: for each nonzero T-submodule E of the quotient field K1 of T, let E∗1 = ∪ {E :K1 JT | J ∈ $\mathcal{F}$∗}, where K1 denotes the quotient field of T and $\mathcal{F}$∗ the localizing system associated to ∗. In this paper we investigate the basic properties of ∗1. Moreover, we show that the map $\varphi$ which associates to a semistar operation ∗ stable and of finite type on R, the semistar operation ∗1 is continuous. Furthermore, we give sufficient conditions for $\varphi$ to be a homeomorphism.