  Arithmetic valuations are intimately connected with the structure of the
ideals of a commutative ring. We show how the generalized idempotent semiring
valuations of Jeffrey and Noah Giansiracusa can be used to make this connection
explicit. Through this generalized valuation theory sufficiently complete
positive valuations give rise to Galois correspondences with the lattice of
ideals of a commutative ring. Up to isomorphism the semiring of ideals of a
commutative ring can be defined as a universal factoring semiring for positive
valuations. We then further show that this valuation theory can formally
connect Joyal's \textit{notion de z\'eros} and universal property of the
spectrum of a ring to arithmetic valuation theory. We show that up to
isomorphism as a coframe, the closed Zariski topology on a spectrum of a
commutative ring can be defined as the universal factoring semiring of
additively and multiplicatively idempotent valuations.
