  Spectral spaces, introduced by Hochster, are topological spaces homeomorphic
to the prime spectra of commutative rings. In this paper we study spectral
spaces in perspective of idempotent semirings which are algebraic structures
receiving a lot of attention due to its several applications to tropical
geometry. We first prove that a space is spectral if and only if it is the
\emph{prime $k$-spectrum} of an idempotent semiring. In fact, we enrich
Hochster's theorem by constructing a subcategory of idempotent semirings which
is antiequivalent to the category of spectral spaces. We further provide
examples of spectral spaces arising from sets of congruence relations of
semirings. In particular, we prove that the \emph{space of valuations} and the
\emph{space of prime congruences} on an idempotent semiring are spectral, and
there is a natural bijection of sets between the two; this shows a stark
difference between rings and idempotent semirings. We then develop several
aspects of commutative algebra of semirings. We mainly focus on the notion of
\emph{closure operations} for semirings, and provide several examples. In
particular, we introduce an \emph{integral closure operation} and a
\emph{Frobenius closure operation} for idempotent semirings.
