Reductions of Ideals in Prufer Rings

Two well-known results, due to Hays, assert that an integral domain is Prufer if and only if every finitely generated ideal is basic, and it is one-dimensional Prufer if and only if every ideal is basic. This research project investigates reductions of ideals in the family of Prufer rings, with the aim to recover and generalize Hays' results to classes of rings with zero-divisors subject to various Prufer conditions. We provide original examples in zero-divisor environments, where we determine the reductions of ideals, with the aim to illustrate the obtained results and delimit their respective scopes.