Reductions and cores of ideals in trivial ring extensions

Abstract

This paper contributes to the study of reductions and cores of ideals in commutative rings. We pursue our investigation initiated in our previous research works on reductions and core, but this time, in settings with zero-divisors. Namely, we study reductions and cores of ideals in various contexts of trivial ring extensions. Section 2 presents preliminary results on reductions and core in generic trivial extensions and Sec. 3 investigates reductions and cores of arbitrary ideals (i.e. not necessarily saturated) in trivial extensions issued from special categories of modules. All results are illustrated with original examples in Sec. 4.