Normal Sally modules of rank one

In this paper, we explore the structure of the normal Sally modules of rank one with respect to an m-primary ideal in a Nagata reduced local ring R which is not necessary Cohen-Macaulay. As an application of this result, when the base ring is Cohen-Macaulay analytically unramified, the extremal bound on the first normal Hilbert coefficient leads to the depth of the associated graded rings G‾ with respect to a normal filtration is at least dim⁡R−1 and G‾ turns in to Cohen-Macaulay when the third normal Hilbert coefficient is vanished.