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Strongly distributive multiplicative hyperrings. (English) Zbl 0716.16023
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.
Reviewer: G.L.Booth

##### MSC:
 16Y99 Generalizations
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##### References:
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