Note on isomorphism theorems of hyperrings. (Q623691)

There are different kinds of hyperrings [\textit{B. Davvaz, V. Leoreanu-Fotea}, Hyperring theory and applications. Palm Harbor: International Academic Press (2007; Zbl 1204.16033)]. In this paper, the authors extend the isomorphism theorems of hyperrings, where the additions and multiplications are hyperoperations. In page 3, the authors give a definition for a multiplicative hyperring \((R,+,\cdot)\) which is different from the usual definition of multiplicative hyperrings. Indeed, they suppose that the strong distributive law holds in a multiplicative hyperring. But in this case, we obtain an ordinary ring instead of a hyperring. Indeed, for all \(a,b\in R\), \(|a\cdot b|=1\). Because, we have \(0=a\cdot 0\), i.e., \(|a\cdot 0|=1\). Also, we have \(0=a\cdot 0=a\cdot(b-b)=a\cdot b-a\cdot b\). Now, suppose that \(x,y\in a\cdot b\). Then, \(x-y\in a\cdot b-a\cdot b\) which implies that \(x-y\in a\cdot 0\) and so \(x=y\).