Let \(R\) be an analytically unramified Noetherian local ring with the maximal ideal \(\mathfrak{m}\) and \(d=\dim R>0\). Let \(I\) be an \(\mathfrak{m}\)-primary ideal of \(R\) and suppose that \(I\) contains a parameter ideal \(Q=(a_1, a_2, \dots, a_d)\) of \(R\) as a reduction. Let \(\ell(M)\) denote the length of an \(R\)-module \(M\) and \(\overline{I^{n+1}}\) denote the integral closure of \(I^{n+1}\) for each \(n\geq 0\). There are integers \(\{\overline{e_i}(I)\}_{0\leq i\leq d}\) such that the equality \[ \ell(R/{\overline{I^{n+1}}})=\sum_{i=0}^d(-1)^{i}\overline{e_i}(I){{n+d-i}\choose{d-i}}, \] holds true for all integers \(n\gg 0\), which is called the normal Hilbert coefficients of \(R\) with respect to \(I\). Let \(\{{e_i}(I)\}_{0\leq i\leq d}\) be the ordinary Hilbert coefficients of \(R\) with respect to \(I\). Let \(\mathcal{R}=R[It]\) and \(T=R[Qt]\) be the Rees algebra of \(I\) and \(Q\), respectively. Let \(\overline{\mathcal{R}}\) and \(\overline{\mathcal{G}}\) denote the integral closure of \(\mathcal{R}\) and the associated graded ring of the normal filtration \(\{\overline{I^n}\}_{n\in\mathbb{Z}}\), respectively. The author defines the normal Sally modules \(\overline{S}=\overline{S}_Q(I)\) of \(I\) with respect to a minimal reduction \(Q\) and also she considers the following four conditions:
\((C_0)\) The sequence \(a_1, a_2, \dots, a_d\) is a \(d\)-sequence in \(R\).
\((C_1)\) The sequence \(a_1, a_2, \dots, a_d\) is a \(d^{+}\)-sequence in \(R\), that is for all integers \(n_1, n_2, \dots, n_d\geq 1\) the sequence \(a_1^{n_1}, a_2^{n_2}, \dots, a_d^{n_d}\) forms a \(d\)-sequence in any order.
\((C_2)\) \((a_1,a_2,\dots,\breve{a_1}, \dots, a_d):_R a_i\subseteq I\) for all \(1\leq i\leq d\).
\((C_3)\) \(\mathrm{depth} R>0\) and \(\mathrm{depth} R>1\) if \(d\geq 2\). The main result of this paper is as follows:
{Theorem 1.1.} Let \(R\) be a Nagata and reduced local ring with the maximal ideal \({\mathfrak{m}}\) and \(d=\dim R>0\). Let \(I\) be an \(\mathfrak{m}\)-primary ideal of \(R\) and suppose that \(I\) contains a parameter ideal \(Q\) of \(R\) as a reduction. Assume that conditions \((C_1), (C_2),\) and \((C_3)\) are satisfied. Then the following are equivalent to each other.
(1) \(\overline{e_1}(I)=e_0(I)+e_1(Q)-\ell(R/{\overline{I}})+1\).
(2) \({\mathfrak{m}}{\overline{S}}=(0)\) and \(\mathrm{rank}_B(\overline{S})=1\), where \(B=T/{\mathfrak{m}}T\).
(3) \(\overline{S}\cong B(-q)\) as graded \(T\)-modules for some integer \(q\geq 1\).
When this is the case
(a) \(\overline{S}\) is a Cohen-Macaulay \(T\)-module.
(b) Put \(t=\mathrm{depth} R\). Then \(\mathrm{depth}\overline{\mathcal{G}}\geq d-1\) when \(t\geq d-1\) and \(\mathrm{depth}\overline{\mathcal{G}}\geq t\) when \(t\leq d-2\).
(c) For all \(n\geq 0\), \(\ell(R/{\overline{I^{n+1}}})=e_0(I){{n+d}\choose{d}}-\{e_0(I)+e_1(Q)-\ell(R/\overline{I})\}{{n+d-1}\choose{d-1}}+ \sum_{i=2}^d(-1)^i\{e_{i-1}(Q)+e_i(Q)\}{{n+d-i}\choose{d-i}}\) if \(n\leq q-1\), and \(\ell(R/{\overline{I^{n+1}}})=e_0(I){{n+d}\choose{d}}-\{e_0(I)+e_1(Q)-\ell(R/\overline{I})+1\}{{n+d-1}\choose{d-1}}+ \sum_{i=2}^d(-1)^i\{e_{i-1}(Q)+e_i(Q)+{{q}\choose{i-1}}\}{{n+d-i}\choose{d-i}}\) if \(n\geq q\). Hence \(\overline{e_i} (I)=e_{i-1}(Q)+e_i(Q)+{{q}\choose{i-1}}\) for all \(2\leq i\leq d\).
As an application of Theorem 1.2, she proves the following result:
{Theorem 1.2.} Let \(R\) be a analytically unramified Cohen-Macaulay local ring with the maximal ideal \({\mathfrak{m}}\), dimension \(d=\dim R > 0\), and \(I\) an \({\mathfrak{m}}\)-primary ideal of \(R\) containing a parameter ideal \(Q\) of \(R\) as a reduction. Assume that \(\overline{e_1}(I)=e_0(I)-\ell(R/\overline{I})+1\). Then the following assertions hold true.
(1) \(\mathrm{depth}\overline{\mathcal{G}}\geq d-1\).
(2) If \(d\geq 3\) and \(\overline{e_3}(I)=0\), then \(\overline{\mathcal{G}}\) is Cohen-Macaulay, \(\overline{e_2}(I)=1\), and the normal filtration has reduction number two.