Finitely generated rings obtained from hyperrings through the fundamental relations

Abstract

In this article, we introduce and analyze a strongly regular relation $\omega^{*}_{\mathcal{A}}$ on a hyperring$R$ such that in a particular case we have $|R/\omega^{*}_{\mathcal{A}}|\leq 2$ or$R/\omega^{*}_{\mathcal{A}}=<\omega^{*}_{\mathcal{A}}(a)>$, i.e., $R/\omega^{*}_{\mathcal{A}}$ is a finite generated ring. Then, by using the notion of $\omega^{*}_{\mathcal{A}}$-parts, we investigate the transitivity condition of $\omega_{\mathcal{A}}$. Finally, we investigate a strongly regular relation $\chi^{*}_{\mathcal{A}}$ on the hyperring $R$ such that $R/\chi^{*}_{\mathcal{A}}$ is a commutative ring with finite generated.