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A class of hyperrings. (English) Zbl 0719.16025

A hyperring is a nonempty set R, together with two hyperoperations, \(\oplus\) and \(\odot\) such that: I. \(<R,\oplus >\) is a hypergroup. II. \(<R,\odot >\) is a semihypergroup. III. For all x,y,z\(\in R\), it holds that \[ (i)\quad x\odot (y\oplus z)\subset (x\odot y)\oplus (x\odot z)\text{ and } (ii)\quad (y\oplus z)\odot x\subset (y\odot x)\oplus (z\odot x). \] This definition is more general than those of M. Krasner [Actes Colloq. d’Algèbre supérieure, C.B.R.M., Bruxelles, 19-22 décembre 1956, 129-206 (1957; Zbl 0085.265)] and R. Rota [Rend. Mat. Appl., VII. Ser. 2, 711-724 (1982; Zbl 0519.16026)]. If only III(i) (resp. III(ii)) is valid, then R is called a left (right) distributive hyperring. Let \((R,+,.)\) be a ring, and let \(P_ 1\) and \(P_ 2\) be two nonempty subsets of R. We define \(xP^*_ 1y=\{x+p_ 1+y:\) \(p_ 1\in P_ 1\}\) and \(xP^*_ 2y=\{xp_ 2y:\) \(p_ 2\in P_ 2\}\). It is shown that if \(P_ 1\neq P_ 2\), \(P_ 1,P_ 2\neq \emptyset\) and \(RP_ 1P_ 2\subset P_ 1\), then \((R,P^*_ 1,P^*_ 2)\) is a left distributive hyperring. Such a hyperring is called a left P-hyperring. Right P-hyperrings are constructed analogously. Various special cases are examined, and in particular it is shown that \((R,P_ 1,P_ 2)\) is a multiplicative hyperring in the sense of Rota if and only if \(P_ 1=\{0\}\). Various elementary properties of P-hyperrings are deduced. Hyperideals of R are defined, and it is shown that if I is a hyperideal of R, then a quotient hyperring R/I may be constructed in a natural way. Furthermore, R/I is a multiplicative hyperring, i.e. the first operation is single-valued.

MSC:

16W99 Associative rings and algebras with additional structure
16Y99 Generalizations
20N20 Hypergroups
16D25 Ideals in associative algebras
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