The w-integral closure of integral domains

Let D be an integral domain with quotient field K, $\overline{D}$ the integral closure of D, X an indeterminate over D, and $N_v=\{f\in D[X]|{(A_f)}_v=D\}$. Let w be the 1-operation on D defined by $I_w=\{x\in K|\text{there is a finitely generated ideal}A\text{such that}{A}^{\mathrm{-1}}=D\text{and}xA\subseteq I\}$, and let $D^w=\{u\in K|u{I}_w\subseteq I_w\text{for some nonzero finitely generated ideal}I\text{of}D\}$. Then $D^w$, called the w-integral closure of D, is an integrally closed overring of D. In this paper, we show that $D^w=\overline{D}{[X]}_{N_v}\cap K$ and $D^w{[X]}_{N_v}=\overline{D}{[X]}_{N_v}$. Using this result, we give several w-integral closure analogs of the integral closure. We also study the w-integral closure of UMT-domains and strong Mori domains.