Let $D$ be an integral domain and $I$ be an ideal of the upper trangular matrix ring $T_{n}(D)$. In this paper, we study the equalizing ideal$$q_{I}=\{A\in T_n(D)|f(A)-f(0)\in I,\forall f\in {\operatorname{Int}}(T_n(D))\}.$$of the integer-valued polynomials over $T_{n}(D)$. 1. IntroductionLet $\mathbb{Z}$ and $\mathbb{Q}$ denote the set of integer and rational numbers, respectively. The classical ring of the integer-valued polynomials (over $\mathbb{Z}$) is defined by $${\operatorname{Int}}(\mathbb{Z}):=\{f\in \mathbb{Q}[x] |f(\mathbb{Z})\subseteq \mathbb{Z}\}.$$ This ring has many interesting properties and has been extensively investigated. For a generalization, let $D$ be an integral domain and let $K$ be the field of fractions of $D$. Then, the ring of integer-valued polynomials over $D$ is defined by $${\operatorname{Int}}(D):=\{f\in K[X] |f(D)\subseteq D\}.$$ The first systematic study of the algebraic properties of ${\operatorname{Int}}(D)$ was done in consecutive 1919 papers of Ostrowski [8] and Polya [9] of the same title. Let $R$ be a ring and $f(x)$ and $g(x)$ be two elements of $R[x]$. Then $(fg)(x)$ denotes the product of $f(x)$ and $g(x)$ in $R[x]$. If $R$ is not a commutative ring and $\alpha\in R$, then $(fg)(\alpha)$ need not equal $f(\alpha)g(\alpha)$. In this case if $f(x)=\sum_{i=1}^{n}a_{i}x^{i}$, then we may express$$(fg)(X)=\sum_{i=1}^{n}a_{i}g(X)X^{i}.\tag{*}$$Let $S$ be a non empty set and let $M_n(S)$ and $T_n(S)$ denote the set of $n\times n$ matrices and upper Triangular matrices with entries from the set $S$, respectively. Let $D$ be an integral domain. In 2012, Werner showed that the set$$Int(M_{n}(D)): = \{f\in M_{n}(K)[X]|f(M_{n}(D))\subseteq M_{n}(D) \}$$with ordinary addition and multiplication $(*)$ is a ring [12, Corollary 1.3]. In 2017, Frisch proved that$$Int(T_{n}(D)) := \{f\in T_{n}(K)[X]|f(T_{n}(D))\subseteq T_{n}(D) \}$$is a ring \cite[3, Theorem 5.4]. Recently, integer-valued polynomials have been considered over some noncommutative rings (see for examples [3, 5, 6, 10]. In [1], Cahen and Chabert introduced the notion of equalizing ideal for a maximal ideal of $D$. In 2015, Cahen and Rissner [2] generalized this notion for an arbitrary ideal of the domain $D$. The authors in [5], introduced and studied the equalizing ideal of every ideal $M_n(\frak{a})$ in ${\operatorname{Int}} M_n(D))$, where $\frak{a}$ is an ideal of $D$. We generalize the notion of equalizing ideal in the integer-valued polynomial ring ${\operatorname{Int}}(T_n(D))$. In fact, for an arbitrary ideal $I$ of $T_n(D)$, we define$$q_{I}=\{A\in T_n(D)|f(A)-f(0)\in I,\forall f\in {\operatorname{Int}}(T_n(D))\}.$$ If $I$ is an ideal of $T_n(D)$ and $A\in T_n(D)$, we set $$\mathcal{J}_{I,A}:=\{f\in{\operatorname{Int}}(T_n(D))|f(A)\in I\}.$$ In this paper, we study the equaizing ideal $q_{I}$ and find some relations between $q_{I}$ and $\mathcal{J}_{I,A}$. 2. Main ResultsThe main results of this paper are stated as follows:Theorem 2.1. Let $I$ be an ideal of $T_n(D)$ and let $A_1,A_2\in T_n(D)$. Then the following statments hold. (1) $q_{I}$ is an ideal of $T_n(D)$. (2) $q_{I}\subseteq I$. (3) If $A_1 - A_2\in q_{I}$, then $\mathcal{J}_{I,A_1}=\mathcal{J}_{I,A_2}$. Theorem 2.2. Let $n>1$ and let $f(X)=B_{k}X^{k}+\cdots+B_{1}X+B_{0}\in{\operatorname{Int}}(T_{n}(D))$. Then, the following statments hold. (1) $B_{0}\in T_{n}(D)$. (2) $(B_{1})_{ij}\in D$ for all $1\leq i\leq j\leq n-1$.The following corollary, which is one of the mail results of this paper, immediatelly follows.Corollary 2.3. Let $\frak{a}$ be an ideal of $D$. Then, we have $$T_{n}(q_{\frak{a}})\subseteq q_{T_{n}(\frak{a})}.$$ Let $R$ be a commutative ring with identity and let $I$ be an ideal of $T_n(R)$. By [4], there are ideals $I_{ij}$ $(i\leq j)$ of $R$ such that\begin{equation*} I=\begin{bmatrix}I_{11} & I_{12} & \ldots &I_{1n}\\0 & I_{22} & \ldots &I_{2n}\\\vdots & \vdots &\ddots & \vdots\\0 & \ldots & 0 &I_{nn}\\\end{bmatrix}\!\!,\end{equation*}where $I_{ii}\subseteq I_{i(i+1)}\subseteq \cdots \subseteq I_{in}$ and $I_{ii}\subseteq I_{(i-1)i}\subseteq \cdots \subseteq I_{1i}$ for all $1\leq i\leq n$. By this notation, we have the following theorem. Theorem 2.4. Let $I$ be an ideal of $T_n(D)$. Then, we have\begin{equation*}q_I=\begin{bmatrix}q_{I_{11}} & I_{12} & \ldots &I_{1n}\\0 & q_{I_{22}} & \ldots &I_{2n}\\\vdots & \vdots &\ddots & \vdots\\0 & \ldots & 0 &q_{I_{nn}}\end{bmatrix}\!\!.\end{equation*} Another main result of this paper is the following corollary.Corollary 2.5. Let $I$ be a nonzero ideal of ideal of $T_{n}(D)$. Then $q_{I}\neq 0.$ 3. ConclusionsLet $\frak{a}$ be an ideal of $D$. It is not always easy to compute the equalizing ideal $q_{T_{n}(\frak{a}))}$ (or $q_{\frak{a}}$). However, it is known that, for every nonzero ideal $\frak{a}$ of a one-dimensional, Noetherian, local domain $D$ with finite residue field, the residue ring $\frac{D}{\frak{a}}$ is finite. In this paper, we prove that the set of distinct ideals of the form $\mathcal{J}_{T_{n}(\frak{a}),A}$ is finite for one-dimensional, Noetherian, local domain $D$ with finite residue field ($A\in T_n(D)$ and $\frak{a}$ be an ideal of $D$). Let $D$ be a local domain with maximal ideal $\frak{m}$ and $a\in D$. As a consequence, we have that, if $D$ is a Noetherian local one-dimensional domain with finite residue field, which is not unibranched, then the set of distinct ideals $\mathcal{J}_{\frak{m},a}$ of ${\operatorname{Int}}(D)$ above the maximal ideal $\frak{m}$ of $D$ is finite (see [1, Proposition V.3.10]).