Southeast Asian Bulletin of Mathematics (2002) 26: 203–213 Southeast Asian Bulletin of Mathematics : Springer-Verlag 2002 On the Operator Semirings of a G-Semiring T.K. Dutta Department of Pure Mathematics, University of Calcutta 35, Ballygunge Circular Road, Calcutta-700019, India S.K. Sardar Department of Mathematics, University of Burdwan, Golapbag, Burdwan-713104, W.B., India AMS Subject Classification (2000): 16Y60, 16Y99 Abstract. In this paper we introduce the notion of operator semirings of a G-semiring to study G-semirings. It is shown that the lattices of all left (right) ideals (two-sided ideals) of a G-semiring and its right (respectively left) operator semiring are isomorphic. This has many applications to characterize various G-semirings. Keywords: G-semiring, Operator semiring of a G-semiring, G-semifield, Noetherian G- semiring, 0-simple G-semiring. 1. Introduction The notion of G-semiring was introduced by M. Murali Krishna Rao [11] as a generalization of G-ring [10, 2] as well as of semiring. He studied [12] G-semirings with left (right) unity, sub-G-semirings, ideals, prime ideals, k-ideals, h-ideals in G- semirings, regular G-semirings etc. In this paper we slightly change the defining conditions of G-semiring of Rao [11] and then introduce the notion of left operator semiring and right operator semiring of a G-semiring as J. Luh [8, 7] did in the theory of G-rings. We show that lattices of all right ideals (two-sided ideals) of a G-semiring and its left operator semiring are isomorphic. Similar result is also obtained for the right operator semiring. Using these results we characterize vari- ous G-semirings. It should be noted that the notion of G in algebra was first introduced [10] by N. Nobusawa as a generalization of ring as well as of Hestenes’ ternary rings [6]. 2. Preliminaries Definition 2.1. Let S and G be two additive commutative semigroups. Then S is called a G-semiring if there exists a mapping S  G  S ! S (images to be denoted