Classical Zariski topology on the prime spectrum of a semimodule

We introduce the classical Zariski topology on the prime spectrum of a semimodule and study this topological space from the viewpoint of spectral spaces. First, we prove that every irreducible closed subset has a generic point in the prime spectrum of a semimodule and that the prime spectrum of a semimodule is a spectral space if it is a Noetherian topological space. Then, we prove that the prime spectrum of a semimodule is a Noetherian topological space if and only if the semimodule satisfies the ascending chain condition on intersection of prime subsemimodules. Also, we show that each subsemimodule in a semimodule satisfying the ascending chain condition on intersection of prime subsemimodules has at most finitely many minimal prime divisors. As consequences, we give positive and complete answers to the Conjecture in [Int. Electron. J. Algebra 4 (2008) 104] and Question 3.5 in [Int. Electron. J. Algebra 4 (2008) 131].