A Fundamental Theorem of Homomorphisms for Semirings

1. Introduction. When studying ideal theory in semirings, it is natural to consider the quotient structure of a semiring modulo an ideal. If 7 is an ideal in a semiring R, the collection {x+l}xeit oí sets x + I={x+i\iEl} need not be a partition of R. Faced with this problem, [2] used equivalence relations to determine cosets relative to an ideal. La Torre successfully established analogues of several well-known isomorphism theorems for rings. However, the methods that Bourne and La Torre used to construct quotient structures proved to be unsuccessful when trying to obtain an exact analogue of the Fundamental Theorem of Homomorphisms. In this paper, the notion of a Ç-ideal will be defined and a construction process will be presented by which one can build the quotient structure of a semiring modulo a Q-ideal. Maximal homomorphisms will be defined and examples of such homomorphisms will be given. Using these notions, the Fundamental Theorem of Homomorphisms will be generalized to include a large class of semirings.