Arithmetic profiles and the prime spectrum of the integer necklace ring

Let R=Nr( Z) be the integer necklace ring and let B=∏n≥1 Z be its ghost product. We study ideal radicals and the ordinary prime spectrum through this embedding. For every ideal I⊆ R, we prove (IB∩ R)²⊆ I. It follows that radicals contract and every prime of R lifts to B. We characterize the coordinatewise gcd profiles of finite ghost families by truncated valuation conditions and show that two ghosts realize every such profile. Consequently every finitely generated ideal of R has arithmetical rank at most two. The profiles give an arithmetic presentation of the lattice of quasicompact Zariski opens and of the spectrum, including specialization and the patch topology. A least compatible multiple construction gives infinitary divisibility criteria for equality and containment of ambient prime contractions. We also classify the primes closed in the necklace coordinate topology and describe all ambient primes above each of them. The resulting account distinguishes the full ordinary spectrum from its coordinate-topology-closed part and relates both to the known positive-characteristic and p-adic constructions.