Reductions and cores of ideals in polynomial rings over integral domains

This paper examines reductions and cores of ideals in polynomial rings over integral domains, with a particular focus on valuation and Prüfer domains. The main objective is to derive explicit formulas for reductions and cores of key ideal classes, such as extended ideals, uppers of prime ideals, and divisorial ideals, emphasizing stability and the basic property. To provide a broader foundation, we first examine reductions and cores in extensions of Prüfer domains, establishing key properties of reductions in extensions that are significant on their own and essential for later sections on polynomial rings. By leveraging results on extensions, we further refine explicit computations of the core in polynomial rings and develop new insights into the structure of ideals in these settings. Throughout the paper, illustrative and original examples reinforce the results and clarify the scope of the underlying assumptions.