Strongly distributive multiplicative hyperrings (Q752817)

A hyperring is a triple \((A,+,\circ)\) where \((A,+)\) is an abelian group, \(a\circ b\) is a subset of A for each a,b\(\in A\), and the following axioms are satisfied \(\forall a,b,c\in A:\) (i) \(a\circ (b\circ c)=(a\circ b)\circ c\) (ii) \((a+b)\circ c\subseteq a\circ c+b\circ c\) (iii) \(a\circ (b+c)\subseteq a\circ b+a\circ c\) (iv) \((-a)\circ b=a\circ (-b)=-(a\circ b)\). If equality holds in (ii) (resp. (iii)) then A is called strongly left (right) distributive. If A is both strongly left and strongly right distributive, then it is called a strongly distributive hyperring. Various results are proved for strongly left and right hyperrings, in particular the following: Let \((A,+,\circ)\) be a strongly left (right) hyperring such that, for any a,b\(\in A| a\circ b| =k>1\). Then \((A,+,\circ)\) is also strongly right (left) distributive. The author then gives attention to strongly distributive hyperrings, and obtains the following result inter alia: Let \((A,+,\circ)\) be a strongly distributive hyperring. If a ring \((R,+,.)\) exists, together with a bijection \(\alpha: R\to A\) such that \(\forall x,y\in R\), (i) \(\alpha (x+y)=\alpha (x)+\alpha (y)\); (ii) \(\alpha\) (x\(\cdot y)\in \alpha (x)\circ \alpha (y)\), then if we denote by S the hyperideal \(0\circ 0\) and \(T=\alpha^{-1}(S)\), T is an ideal of R and the quotient \((R,+,.)/T\) is isomorphic to the ring \((A,+,\circ)/S\). Moreover, it is possible to define in \((A,+)\) a product \(\times\) in order to obtain a ring \((A,+,\times)\) isomorphic to \((R,+,.)\) through \(\alpha\). Furthermore, \((A,+,\times)/S\) will be isomorphic to \((R,+,.)/T\) and \(\forall a,b\in A\), \(a\circ b=a\times b+S\). A similar result is obtained under a somewhat different hypothesis.