Maximal ideals in semirings of continuous functions

The author investigates the conditions on a topological space $X$ and a topological semiring $S$ of the existence of a canonical homeomorphism between the maximal ideal space of the semiring $C(X,S)$ and the Tikhonov product of $X$ and the maximal ideal space of the semiring $S$. The generalized zero sets and the properties of continuous function semirings are considered. An analog of Gelfand–Kolmogorov theorem is obtained for maximal ideals of semirings of continuous functions with values in topological semiskewfields.