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The structure spaces, endowed with the hull kernel topology, of maximal regular congruences, which are prime on a [Formula: see text]-semiring as well as the structure space of prime congruences on a [Formula: see text]-semiring, have been studied via operator semirings. This study has been used to obtain some important results of the structure space of [Formula: see text] of nonpositive real valued continuous functions over a topological space [Formula: see text]. It has been found that the structure spaces of the semiring [Formula: see text] of nonnegative real valued continuous functions and the [Formula: see text]-semiring [Formula: see text] are homeomorphic. Moreover it has been shown that the structure space of [Formula: see text] is the Stone–Čech compactification of [Formula: see text], where [Formula: see text] is a Tychonoff space. Furthermore, the [Formula: see text]-semiring analogue of the ‘Banach–Stone Theorem’ has been obtained.