On $w$-Hilbert domains

In this paper, we introduce the notion of a $w$-Hilbert domain and investigate its basic properties. More precisely, we explore its relationship with Hilbert domains, strong Mori domains, and UMT domains by providing various examples using $D+M$ constructions. Furthermore, we establish necessary and sufficient conditions for the polynomial ring and the Anderson ring to be $w$-Hilbert domains, and compare the $w$-dimension of the polynomial ring with that of its base ring.