Extriangulated ideal quotients and Gabriel-Zisman localizations

Let $(\mathcal B,\mathbb{E},\mathfrak{s})$ be an extriangulated category and $\mathcal S$ be an extension closed subcategory of $\mathcal B$. In this article, we prove that the Gabriel-Zisman localization $\mathcal B/\mathcal S$ can be realized as an ideal quotient inside $\mathcal B$ when $\mathcal S$ satisfies some mild conditions. The ideal quotient is an extriangulated category. We show that the equivalence between the ideal quotient and the localization preserves the extriangulated category structure. We also discuss the relations of our results with Hovey twin cotorsion pairs and Verdier quotients.