ABSTRACT
Let R be an integral domain with quotient field K. A prime ideal P of R is called strongly prime if x, y ∈ K and xy ∈ P imply that x ∈ P or y ∈ P [HH1]. An ideal I of R is called strongly radical if whenever x ∈ K satisfies xn ∈ I for some n ≥ 1, then x ∈ I [AA]. Following [SS], an integral domain R is called rooty if each radical ideal of R is strongly radical (equivalently, each prime ideal of R is strongly radical [AP, Theorem 1.8]). Thus valuation domains are rooty domains [AP, Remark 1.9], and we use this fact in the section 2. R is called a pseudo-valuation domain (PVD) if each prime ideal of R is strongly prime [HH1]. It is known that R is a PVD if and only if (K − R) ∪ U(R) is multiplicatively closed where U(R) is the group of units of R [AA, Theorem 1.2]. On the other hand, R is a valuation domain if and only if K − R is multiplicatively closed. Note that a strongly prime ideal is strongly radical, and so PVDs are rooty. In fact, PVDs are characterized as quasilocal domains (R,M) with the property that (M : M) is a valuation domain with maximal ideal M [HH1]. On the other hand, (R,M) is a quasi-local rooty domain if and only if (M : M) is root-closed with radical ideal M . Also, if R is a quasi-local domain that is not rooty, then R is root-closed [AP, Theorem 2.1].
