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


16Y99 Generalizations
Full Text: DOI


[1] CORSINI, P.: Hypergroupes et groupes ordonn?s. Rend. Sem. Mat. Univ. Padova, 48 (1973) · Zbl 0295.06016
[2] CORSINI, P.: I prolegomeni alla teoria degli ipergruppi. Quad. Ist. Mat. Inf. Sist. Univ. Udine
[3] MITTAS, J.: Hyperanneaux et certaines de leurs propriet?s. C.R. Acad. Sci. Paris, A (1969) · Zbl 0205.05301
[4] PROCESI, R. et ROTA, R.: Le spectre premier d’un hyperanneau multiplicatif. Atti Conv. ?Ipergruppi, altre strutture multivoche e loro applicazioni? Udine (1985)
[5] ROTA, R.: Sugli iperanelli moltiplicativi. Rend. Mat. 4 (1982) v.2 s.VII · Zbl 0519.16026
[6] SCAFATI TALLINI, M.: A-iper-raoduli e spazi iper-vettoriali. Riv. Mat. Pura e Appl., Univ. Udine 3 (1988)
[7] VOUGIOUKLIS, T.: Representations of hypergroups. Hypergroup algebra. Atti Conv. ?Ipergruppi, altre strutture multivoche e loro applicazioni?, Udine (1985)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.