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Semiring Unions of a Ring and a Half-Body

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Semiring Unions of a Ring and a Half-Body ISSN 1066-369X, Russian Mathematics (Iz. VUZ), 2008, Vol. 52, No. 12, pp. 65–68. c Allerton Press, Inc., 2008. Original Russian Text c M.A. Lukin, 2008, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2008, No. 12, pp. 76–80. Semiring Unions of a Ring and a Half-Body M. A. Lukin1* 1Vyatka State University of Humanities, ul. Krasnoarmeiskaya 26, Kirov, 610002 Russia Received May 03, 2007; in final form, October 09, 2008 Abstract—We describe the structure of semirings which are unions of a ring and a half-body. DOI: 10.3103/S1066369X08120104 Key words and phrases: ring, semiring, half-body, semiring disjoint union. Brief communication In the theory of semirings the following classes of semirings play an important role: (associative) rings, distributive lattices with zero, and half-bodies with zero. A natural problem is the investigation of semirings reducible to these three types. In [1] (proposition 12.15) one describes semirings which are subdirect products of a certain ring and a certain bounded distributive lattice. In paper [2] one studies Abelian regular positive semirings S, whose structure essentially depends on a distributive lattice of idempotents L(S) and a half-body of invertible elements U(S). This paper follows [3], it is dedicated to the investigation of semirings which represent the union of their two subsemirings, one of which is isomorphic to a ring, and other one is isomorphic to a half-body withzero.Wegiveadefinition of the semiring disjoint union S = R∪˙ U of a ring R and a half-body U and adduce several examples. The case of the semiring union is reduced to the semiring disjoint union. Further we study the structure of the ring R and the half-body U which enter in S = R∪˙ U; based on them, we construct semirings R∪˙ U themselves. Finally, we solve the question about the uniqueness of S = R∪˙ U for given R and U. 1. Preliminary information. A semiring is an algebraic structure S;+, · , 0 such that S;+, 0 is a commutative monoid, S; · is a semigroup, and in S the identities a(b + c)=ab + ac, (a + b)c = ac + bc,anda0=0a =0are fulfilled [1]. A division semiring different from a ring is said to be a half-body with zero.Excluding0 from a half- body S, we obtain the structure S;+, ·;wecallitahalf-body. A half-body U is said to be reducible, if the implication a + c = b + c ⇒ a = b is fulfilled in it. Let S be a certain semiring with unit 1. We denote by r(S) the set of all elements of S which have the opposite one, we do by U(S) the set of all invertible elements of the semiring S.Evidently,r(S) isaring with respect to operations in S. Definition. We call a semiring S with a unit the semiring disjoint union of a ring R and a half-body U; we denote it by R∪˙ U, if it is the union of its nonintersecting subsets r(S) and U(S),wherer(S) is a ring R, U(S) is a half-body U. Example 1.1. Let P be a field and let T be its certain subhalf-body (without zero), let Mn(P ) stand for the ring of all quadratic matrices of the nth order over the field P . The union of the set R of matrices with zero coordinates on the main diagonal and above it and the set U of matrices with zero coordinates above the main diagonal and coordinates from T on the main diagonal with matrix operations Mn(P ) is the semiring disjoint union R∪˙ U. *E-mail: [email protected]. 65.
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