Recently, many scholars paid attention to the significant work has been done in studying the local maps. Larson [1] initially considered local maps in his examination of reflexivity and interpolation for subspaces of $\mathcal {B}(\mathcal {H})$, where $\mathcal {H}$ is a Hilbert space. The notion of local derivations (resp., local automorphisms) was introduced independently by Larson and Sourour [2] and Kadison [3] (resp., Larson and Sourour [2]). Recall that a linear map $\delta$ from an algebra $\mathcal{A}$ into itself is called a local derivation (resp., local automorphism) if for every $a\in \mathcal{A}$, there exists a derivation (resp., an automorphism) $\delta_{a}$ of $\mathcal{A}$, depending on $a$, such that $\delta(a)=\delta_{a}(a)$. If every local derivation (resp., local automorphism) of an algebra is a derivation (resp., an automorphism), then we can say that the derivations (resp., automorphisms) of those structures are, in a certain sense, completely determined by their local actions.

Local derivations, local automorphisms and other local maps have been studied in a variety of contexts. Larson and Sourour [2] showed that every local derivation (resp., every surjective linear local automorphism) on $\mathcal{B}(\mathcal{H})$, the algebra of all bounded linear operators on a Banach space $\mathcal{H}$, is a derivation (an automorphism). Zhao, Yao and Wang [4] proved that every local Jordan derivation (resp., local Jordan automorphism) of upper triangular matrix algebra is an inner derivation (resp., a Jordan automorphism). Other work on the description of the local derivations or local automorphisms on operator algebras can be found in [5-7]. In those articles all local derivations or local automorphisms are actually global derivations or automorphisms. A nontrivial local derivation on an operator algebra was found by Crist in [8]. Crist [9] showed that any linear local automorphism of a finite dimensional CSL algebra $\mathcal{A}$ is either an automorphism or can be factored as an automorphism and the transpose of a self-adjoint summand of $\mathcal{A}$.

The algebra $T_{n}(R)$ of all upper triangular matrices over a commutative ring $R$ is an interesting topic for many researchers. Significant research has been done in studying various linear maps of $T_{n}(R)$. In 1990, Kezlan [10] showed that every $R$-algebra automorphism of $T_{n}(R)$ is inner. Cao [11] and Wang and You [12] gave a description of the Lie automorphisms of $T_{n}(R)$. Tang, Cao and Zhang [13] determined all Jordan isomorphisms of $T_{n}(R)$. Wang and Yu [14] determined the derivations of any Lie subalgebra of the general linear Lie algebra containing $T_{n}(R)$. In this paper, we regard $T_{n}(R)$ as a Lie algebra and we shall study the local automorphisms and local derivations of $T_{n}(R)$.

Let $R$ be a commutative ring with identity, $R^{*}$ the group of invertible elements of $R$. In the following of this paper, we use $T_{n}(R)$ (resp., $D_{n}(R)$) denote the Lie algebra of all upper triangular (resp., diagonal) $n$ by $n$ matrices over $R$, $T_{n}^{*}(R)$ the set of all invertible elements in $T_{n}(R)$. We denote by $\textbf{n}$ the subalgebra of $T_{n}(R)$ consisting of all strictly upper triangular matrices. Let $e$ be the identity matrix of $T_{n}(R)$, $e_{i, j}$ the matrix with 1 at the position $(i, j)$ and zero elsewhere for $1\leq i, j\leq n$. For $x\in T_{n}(R)$, denote by $x^{t}$ the transpose of $x$. Let $\mathcal{S}_{k}=\bigg\{\left(\begin{array}{cc}0&\\ &x\end{array}\right)\in T_{n}(R)\mid x\in T_{n-k}(R)\bigg\}$, $k=1, 2, \cdots, n-1$. Obviously, each $\mathcal{S}_{k}$ is a subalgebra of $T_{n}(R)$.

Cao [11] and Wang and You [12] gave an explicit description of the automorphisms of $T_{n}(R)$, respectively. For convenience of the proof of the main result in this section, we give another description of the automorphisms of $T_{n}(R)$ by the following lemma. Before giving the lemma, let us introduce some standard automorphisms of $T_{n}(R)$ as follows. In this section, 2 is an unit in $R$.

(A) Inner automorphisms

Let $a\in T_{n}^{*}(R)$, the map $\theta_{a}: x\mapsto axa^{-1}$ for all $x\in T_{n}(R)$ is an automorphism of $T_{n}(R)$, which is called an inner automorphism.

(B) Central automorphisms

Regarding $R$ as an abelian Lie algebra. Let

$ F=\{f\in\mbox{ Hom}_{R}(T_{n}(R), R)\mid 1+f(e)\in R^{*}\}. $

For $f\in F$, we define a map $\eta_{f}: x\mapsto x+f(x)e$ for all $x\in T_{n}(R)$. It can be checked that $\eta_{f}$ is an automorphism of $T_{n}(R)$, which is called a central automorphism. Since $\eta_{f}(\textbf{n})=\textbf{n}, f(y)=0$ for any $y\in \textbf{n}$.

(C) Graph automorphisms

Set $r=e_{1n}+e_{2, n-1}+\cdots+e_{n-1, 2}+e_{n1}$. It is clear that $r^{2}=e$ and $r^{t}=r$. Let $\Upsilon$ be the set of all idempotents in $R$. For $\varepsilon\in \Upsilon$, it is easy to check that the map $w_{\varepsilon}: x\mapsto \varepsilon x-(1-\varepsilon)rx^{t}r$ is an automorphism of $T_{n}(R)$. We call $w_{\varepsilon}$ a graph automorphism.

Lemma 2.1(the main theorem of [11] and [12])  Let $\psi$ be an automorphism of $T_{n}(R)$. Then there exist an inner automorphism $\theta_{a}$, a graph automorphism $w_{\varepsilon}$ and a central automorphism $\eta_{f}$ of $T_{n}(R)$ such that $\psi=\theta_{a}w_{\varepsilon}\eta_{f}$ for $n\geq 3$; $\psi=\theta_{a}\eta_{f}$ when $n=2$; $\psi=\eta_{f}$ for $n=1$.

The following lemma is obvious.

Lemma 2.2  Let $\theta$ be an inner automorphism of $T_{n}(R)$. Then $\theta(E)=\theta(E)^{2}$ for every idempotent $E$ in $T_{n}(R)$.

We will prove our main result in this section via the following lemmas.

Lemma 2.3  Let $\varphi$ be a local automorphism of $T_{n}(R)$ ( $n\geq 3$). If $\varphi(e_{11})=e_{11}$, then we may find an inner automorphism $\theta=\prod_{j=2}^{n}\theta_{b_{j}}$ and a central automorphism $\eta_{f}$ such that $\eta_{f}^{-1}\theta^{-1}\varphi(e_{ii})=e_{ii}$ for $i=1, 2, \cdots, n$.

Proof  For $e_{ii}\in T_{n}(R)$, since $\varphi$ is a local automorphism, there exists an automorphism $\varphi_{e_{ii}}$, depending on $e_{ii}$, such that $\varphi(e_{ii})=\varphi_{e_{ii}}(e_{ii})$. By Lemma 2.1, we know there exist $\varepsilon_{i}\in \Upsilon$, $f_{i}\in F$ and $a_{i}\in T_{n}^{*}(R)$ such that

$ \begin{eqnarray} \varphi(e_{ii})=\varphi_{e_{ii}}(e_{ii})=\theta_{a_{i}} w_{\varepsilon_{i}}\eta_{f_{i}}(e_{ii}). \end{eqnarray} $

(2.1)

In the following, we first prove that $\varepsilon_{i}=1$ for $i=2, 3, \cdots, n$ in (2.1).

From (2.1) we get

$ \begin{eqnarray} &&\varphi(e_{11}+e_{ii})=e_{11}+\varphi(e_{ii})\nonumber\\ &\equiv&e_{11}+\varepsilon_{i}e_{ii}+(2\varepsilon_{i}-1)f_{i}(e_{ii})e -(1-\varepsilon_{i})e_{n+1-i, n+1-i}\ {\mbox{mod}}\ \textbf{n}. \end{eqnarray} $

(2.2)

On the other hand, we have $\varphi(e_{11}+e_{ii})=\varphi_{e_{11}+e_{ii}}(e_{11}+e_{ii})$, where $\varphi_{e_{11}+e_{ii}}$ is an automorphism depending on $e_{11}+e_{ii}$. By Lemma 2.1, we have $\varphi_{e_{11}+e_{ii}}=\theta_{a_{i}^{'}} w_{\varepsilon_{i}^{'}}\eta_{f_{i}^{'}}$ for some $a_{i}^{'}\in T_{n}^{*}(R), \ \varepsilon_{i}^{'}\in \Upsilon$ and $f_{i}{'}\in F$, so

$ \begin{eqnarray} &&\varphi(e_{11}+e_{ii})=\theta_{a_{i}^{'}} w_{\varepsilon_{i}^{'}}\eta_{f_{i}^{'}}(e_{11}+e_{ii})\nonumber\\ &\equiv&\varepsilon_{i}^{'}(e_{11}+e_{ii})+(2\varepsilon_{i}^{'}-1) f_{i}^{'}(e_{11}+e_{ii})e -(1-\varepsilon_{i}^{'})(e_{nn}+e_{n+1-i, n+1-i})\ {\mbox{mod}}\ \textbf{n}. \end{eqnarray} $

(2.3)

From (2.2) and (2.3), we have $\varepsilon_{i}=\varepsilon^{'}_{i}=1$ for $i=2, 3, \cdots, n$. That is to say

$ \begin{eqnarray}\varphi(e_{ii})=\theta_{a_{i}} \eta_{f_{i}}(e_{ii}), i=2, 3, \cdots, n.\end{eqnarray} $

(2.4)

Next we use induction to prove that there exists an inner automorphism $\theta$ such that $\theta^{-1}\varphi(e_{ii})=e_{ii}+f_{i}(e_{ii})e$ for $i=2, \cdots, n$, and $\theta^{-1}\varphi(e_{11})=e_{11}$. Let $i=2$ in (2.2) and (2.3), we have

$ \begin{eqnarray}f_{2}(e_{22})=f^{'}_{2}(e_{11}+e_{22}).\nonumber\end{eqnarray} $

So

$ \begin{eqnarray} \varphi(e_{11}+e_{22}) =\theta_{a_{2}^{'}} \eta_{f_{2}^{'}}(e_{11}+e_{22}) =\theta_{a_{2}^{'}}(e_{11}+e_{22})+f_{2}(e_{22})e. \end{eqnarray} $

(2.5)

On the other hand, by (2.4) we have

$ \begin{eqnarray} \varphi(e_{11}+e_{22})&=&e_{11}+\varphi(e_{22}) =e_{11}+\theta_{a_{2}}(e_{22})+f_{2}(e_{22})e. \end{eqnarray} $

(2.6)

(2.5) and (2.6) imply that $\theta_{a_{2}^{'}}(e_{11}+e_{22})=e_{11}+\theta_{a_{2}}(e_{22}).$ The idempotence of $e_{11}+e_{22}$ shows that the image of it under $\varphi$ is also idempotent. So $\theta_{a_{2}}(e_{22})\in \mathcal{S}_{1}.$ Suppose $a_{2}=(a_{ij}^{(2)})_{n\times n}$. Let $b_{2}=(b_{ij}^{(2)})_{n\times n}$, where $b_{11}^{(2)}=a_{11}^{(2)}, b_{ij}^{(2)}=a_{ij}^{(2)}$ for $2\leq i\leq j\leq n$, and $b_{1j}^{(2)}=0$ for $2\leq j\leq n$. Then $\theta_{b_{2}}(e_{22})=\theta_{a_{2}}(e_{22}).$ So

$ \theta_{b_{2}}^{-1}\varphi(e_{22})=e_{22}+f_{2}(e_{22})e \ \ \ {\mbox{and}}\ \ \ \theta_{b_{2}}^{-1}\varphi(e_{11})=e_{11}. $

Denote $\theta_{b_{2}}^{-1}\varphi$ by $\varphi_{1}$.

By induction we assume that there are $\theta_{b_{j}}, j=3, 4, \cdots, k-1$ such that

$\begin{eqnarray*} (\prod\limits_{j=3}^{k-1}\theta_{b_{j}})^{-1}\varphi_{1}(e_{11})=e_{11}\end{eqnarray*} $

and $(\prod_{j=3}^{k-1}\theta_{b_{j}})^{-1}\varphi_{1}(e_{ii})=e_{ii}+f_{i}(e_{ii})e$, where $f_{i}\in F, i=2, 3, \cdots, k-1.$ Denote $(\prod_{j=3}^{k-1}\theta_{b_{j}})^{-1}\varphi_{1}$ by $\varphi_{k-2}$. By (2.4), we know there exist some $u\in T_{n}^{*}(R)$ and $f_{k}\in F$ such that

$ \varphi_{k-2}(e_{kk})=\theta_{u}(e_{kk}+f_{k}(e_{kk})e)=\theta_{u}(e_{kk})+f_{k}(e_{kk})e. $

So

$ \begin{eqnarray} &&\varphi_{k-2}(e_{11}+\cdots+e_{k-1, k-1}+e_{kk})\nonumber\\ &=&e_{11}+\cdots+e_{k-1, k-1}+(f_{2}(e_{22})+ \cdots+f_{k}(e_{kk}))e+\theta_{u}(e_{kk})\end{eqnarray} $

(2.7)

$\begin{eqnarray} &&\equiv&e_{11}+\cdots+e_{kk}+(f_{2}(e_{22})+ \cdots+f_{k}(e_{kk}))e\ {\mbox{mod}}\ \textbf{n}. \end{eqnarray} $

(2.8)

On the other hand, since $\varphi_{k-2}$ is a local automorphism, there exists an automorphism $\psi=\theta_{t}w_{\alpha}\eta_{\sigma}$, where $t\in T_{n}^{*}(R), \ \alpha\in \Upsilon$ and $\sigma\in F$, depending on $e_{11}+\cdots+e_{kk}$, such that

$ \begin{eqnarray} \varphi_{k-2}(e_{11}+\cdots+e_{kk}) &=&\psi(e_{11}+\cdots+e_{kk})\nonumber\\ &\equiv&\alpha(e_{11}+\cdots+e_{kk})+(2\alpha-1)\sigma(e_{11}+\cdots+e_{kk})e\nonumber\\ &-&(1-\alpha)(e_{nn}+\cdots+e_{n+1-k, n+1-k})\ {\mbox{mod}}\ \textbf{n}. \end{eqnarray} $

(2.9)

By (2.8) and (2.9) we get

$ \begin{eqnarray} \varphi_{k-2}(e_{11}+\cdots+e_{kk}) &=&\theta_{t}(e_{11}+\cdots+e_{kk})+(f_{2}(e_{22})+ \cdots+f_{k}(e_{kk}))e. \end{eqnarray} $

(2.10)

From (2.7) and (2.10), we have $\theta_{t}(e_{11}+\cdots+e_{kk})=e_{11}+\cdots+e_{k-1, k-1}+\theta_{u}(e_{kk}).$ Since $e_{11}+\cdots+e_{kk}$ is idempotent, then by Lemma 2.2, we get

$ [e_{11}+\cdots+e_{k-1, k-1}+\theta_{u}(e_{kk})]^{2}=e_{11}+\cdots+e_{k-1, k-1}+\theta_{u}(e_{kk}), $

which means that $\theta_{u}(e_{kk})\in \mathcal{S}_{k-1}.$ Suppose $u=(u_{ij})_{n\times n}$. Let $b_{k}=(b_{ij}^{(k)})_{n\times n}$, where $b_{ii}^{(k)}=u_{ii}, \ i=1, 2, \cdots, k-1, b_{ij}^{(k)}=u_{ij}$ for $k\leq i\leq j\leq n$, and $b_{ts}^{(k)}=0$ for $t=1, 2, \cdots, k-1, t< s\leq n$. By calculating, we have $\theta_{b_{k}}(e_{kk})=\theta_{u}(e_{kk}).$ So

$ \begin{eqnarray} \theta_{b_{k}}^{-1}\varphi_{k-2}(e_{11})&=&e_{11}, \nonumber\\ \theta_{b_{k}}^{-1}\varphi_{k-2}(e_{ii})&=&e_{ii}+f_{i}(e_{ii})e \ {\mbox{for}}\ i=2, \cdots, k.\nonumber \end{eqnarray} $

When $k=n$, let $\theta=\prod_{j=2}^{n}\theta_{b_{j}}$. Then $\theta^{-1}\varphi(e_{11})=e_{11}$, and $\theta^{-1}\varphi(e_{ii})=e_{ii}+f_{i}(e_{ii})e$ for $i=2, \cdots, n$.

Let $f$ be an $R-$linear map satisfying $f(e_{11})=0, f(e_{ii})=f_{i}(e_{ii})$ for $i=2, \cdots, n$ and $f(e_{ij})=0$ for $1\leq i< j\leq n$. Then $f\in $Hom$_{R}(T_{n}(R), R)$. It is easy to check that $1+f(e)\in R^{*}$. Thus $f\in F$ and $\eta_{f}^{-1}\theta^{-1}\varphi(e_{ii})=e_{ii}\ {\mbox{for}}\ i=1, 2, \cdots, n.$

Lemma 2.4  Let $\varphi$ be a local automorphism of $T_{n}(R)$ satisfying $\varphi(e_{ii})=e_{ii}$ for $i=1, 2, \cdots, n$. Then $\varphi(e_{ij})=a_{ij}e_{ij}$, where $1\leq i < j\leq n$ and $a_{ij}\in R^{*}$.

Proof  For $1\leq i < j\leq n$, there exists an automorphism $\varphi_{e_{ii}+e_{jj}}$ which agree with $\varphi$ at $e_{ii}+e_{ij}$. By Lemma 2.1, we know there exist $\beta_{ij}\in \Upsilon$, $\tau_{ij}\in F$ and $u_{ij}\in T_{n}^{*}(R)$ such that $\varphi_{e_{ii}+e_{ij}}=\theta_{u_{ij}}w_{\beta_{ij}}\eta_{\tau_{ij}}.$ So

$ \begin{eqnarray} &&\varphi(e_{ii}+e_{ij})=\theta_{u_{ij}}w_{\beta_{ij}}\eta_{\tau_{ij}}(e_{ii}+e_{ij})\nonumber\\ &\equiv&\beta_{ij}e_{ii}+(2\beta_{ij}-1)\tau_{ij}(e_{ii}+e_{ij})e-(1-\beta_{ij})e_{n+1-i, n+1-i}\ {\mbox{mod}}\ \textbf{n}. \end{eqnarray} $

(2.11)

On the other hand,

$ \begin{eqnarray} \varphi(e_{ii}+e_{ij})=e_{ii}+\varphi(e_{ij})\equiv e_{ii}\ {\mbox{mod}}\ \textbf{n}. \end{eqnarray} $

(2.12)

So $\beta_{ij}=1$ and $\tau_{ij}(e_{ii}+e_{ij})=0$ follow from (2.11) and (2.12). Thus

$ e_{ii}+\varphi(e_{ij})=\varphi(e_{ii}+e_{ij})=\theta_{u_{ij}}(e_{ii}+e_{ij}). $

Similarly, there exists some $h_{ij}\in T_{n}^{*}(R)$ such that

$ e_{jj}+\varphi(e_{ij})=\varphi(e_{jj}+e_{ij})=\theta_{h_{ij}}(e_{jj}+e_{ij}). $

Since $e_{ii}+e_{ij}$ and $e_{jj}+e_{ij}$ are idempotents, by Lemma 2.2, we know that the image of them under $\varphi$ are also idempotent, which imply that $\varphi(e_{ij})=a_{ij}e_{ij}$ for some $a_{ij}\in R$. Clearly, $a_{ij}\in R^{*}$.

Lemma 2.5  Let $\varphi$ be a local automorphism of $T_{n}(R)$ satisfying $\varphi(e_{ii})=e_{ii}$ for $i=1, 2, \cdots, n$. Then there exists an inner automorphism $\theta_{d}$ such that $\theta_{d}\varphi(e_{i, i+1})=e_{i, i+1}$ for $i=1, 2, \cdots, n-1$, and $\theta_{d}\varphi(e_{ii})=e_{ii}$ for $i=1, 2, \cdots, n$.

Proof  By Lemma 2.4, we have $\varphi(e_{i, i+1})=a_{i, i+1}e_{i, i+1}$ with $a_{i, i+1}\in R^{*}$. Let

$ d={\mbox{diag}}(1, a^{-1}_{12}, (a_{12}a_{23})^{-1}, \cdots, (a_{12}a_{23}\cdots a_{n-1, n})^{-1}). $

Then $\theta_{d}^{-1}\varphi(e_{i, i+1})=e_{i, i+1}$ for $i=1, 2, \cdots, n-1$, and $\theta_{d}^{-1}\varphi(e_{ii})=e_{ii}$ for $i=1, 2, \cdots, n$.

Lemma 2.6  Let $\varphi$ be a local automorphism of $T_{n}(R)$. If $\varphi(e_{ii})=e_{ii}$ for $i=1, 2, \cdots, n$, and $\varphi(e_{i, i+1})=e_{i, i+1}$ for $i=1, 2, \cdots, n-1$, then for any $e_{i, i+k}\in T_{n}(R)$, we have

$ \varphi(e_{i, i+k})=e_{i, i+k}. $

Proof  We will prove this lemma by induction on $k\ (k\geq 2)$. When $k=2$, since $\varphi$ is a local automorphism, we have

$ \varphi(e_{i, i+1}+e_{i+1, i+2}+e_{i, i+2}+e_{i+1, i+1})=\phi_{i}^{(2)}(e_{i, i+1}+e_{i+1, i+2}+e_{i, i+2}+e_{i+1, i+1}), $

where $\phi_{i}^{(2)}$ is an automorphism corresponding to $e_{i, i+1}+e_{i+1, i+2}+e_{i, i+2}+e_{i+1, i+1}$. By Lemma 2.1, we know there exist $\gamma_{i}^{(2)}\in \Upsilon$, $\sigma_{i}^{(2)}\in F$ and $x_{i}^{(2)}\in T_{n}^{*}(R)$ such that $\phi_{i}^{(2)}=\theta_{x_{i}^{(2)}}w_{\gamma_{i}^{(2)}}\eta_{\sigma_{i}^{(2)}}$. So

$ \begin{eqnarray} &&\varphi(e_{i, i+1}+e_{i+1, i+2}+e_{i, i+2}+e_{i+1, i+1})\nonumber\\ &\equiv&\gamma_{i}^{(2)}e_{i+1, i+1}-(1-\gamma_{i}^{(2)})e_{n-i, n-i}+(2\gamma_{i}^{(2)}-1)\sigma_{i}^{(2)}(e_{i+1, i+1})e\ {\mbox{mod}}\ \textbf{n}. \end{eqnarray} $

(2.13)

On the other hand, by Lemma 2.4, we have

$ \begin{eqnarray} &&\varphi(e_{i, i+1}+e_{i+1, i+2}+e_{i, i+2}+e_{i+1, i+1})\nonumber\\ &=&e_{i, i+1}+e_{i+1, i+2}+e_{i+1, i+1}+a_{i, i+2}e_{i, i+2} \equiv e_{i+1, i+1}\ {\mbox{mod}}\ \textbf{n}. \end{eqnarray} $

(2.14)

From (2.13) and (2.14), we have $\gamma_{i}^{(2)}=1$ and $\sigma_{i}^{(2)}(e_{i+1, i+1})=0$. So

$ \theta_{x_{i}^{(2)}}(e_{i, i+1}+e_{i+1, i+2}+e_{i, i+2}+e_{i+1, i+1})=e_{i, i+1}+e_{i+1, i+2}+e_{i+1, i+1}+a_{i, i+2}e_{i, i+2}. $

The idempotence of $e_{i, i+1}+e_{i+1, i+2}+e_{i, i+2}+e_{i+1, i+1}$ and Lemma 2.2 impliy that $a_{i, i+2}=1$. So $\varphi(e_{i.i+2})=e_{i.i+2}$.

By induction we assume that $\varphi(e_{i, i+m})=e_{i, i+m}$ for $m=2, \cdots, k-1$. For $e_{i, i+1}+e_{i+1, i+k}+e_{i, i+k}+e_{i+1, i+1}$, similar to the case $k=2$, we can get that there exists some $x_{i}^{(k)}\in T_{n}^{*}(R)$ such that

$ \theta_{x_{i}^{(k)}}(e_{i, i+1}+e_{i+1, i+k}+e_{i, i+k}+e_{i+1, i+1})=e_{i, i+1}+e_{i+1, i+k}+e_{i+1, i+1}+a_{i, i+k}e_{i, i+k}. $

Also by the idempotence of $e_{i, i+1}+e_{i+1, i+k}+e_{i, i+k}+e_{i+1, i+1}$ and Lemma 2.2, we can prove that $a_{i, i+k}=1$. That is to say $\varphi(e_{i, i+k})=e_{i, i+k}$ for any $e_{i, i+k}\in T_{n}(R)$.

Theorem 2.1  Let $R$ be a commutative ring with identity 1 and unit 2, $T_{n}(R)$ the Lie algebra consisting of all upper triangular $n\times n$ matrices over $R$. Then every local automorphism $\varphi$ of $T_{n}(R)$ is an automorphism.

Proof  Let $\varphi$ be a local automorphism of $T_{n}(R)$. When $n\geq 3$, for $e_{11}\in T_{n}(R)$, by the definition of $\varphi$, there exists an automorphism $\varphi_{e_{11}}$, depending on $e_{11}$, such that $\varphi(e_{11})=\varphi_{e_{11}}(e_{11})$. So $\varphi_{e_{11}}^{-1}\varphi(e_{11})=e_{11}$. Obviously, $\varphi_{e_{11}}^{-1}\varphi$ is also a local automorphism of $T_{n}(R)$. By Lemmas 2.3, 2.5 and 2.6, there are $\eta_{f}^{-1}, \theta^{-1}$ and $\theta_{d}^{-1}$ such that

$ \theta_{d}^{-1}\eta_{f}^{-1}\theta^{-1}\varphi_{e_{11}}^{-1}\varphi(e_{ij})=e_{ij}\ {\mbox{for}}\ 1\leq i\leq j\leq n, $

which mean that $\varphi=\varphi_{e_{11}}\theta\eta_{f}\theta_{d}.$ So $\varphi$ is an automorphism.

When $n=1$, suppose that $\varphi(1)=a$, then for any $x\in T_{1}(R)=R$ we have $\varphi(x)=x\varphi(1)=xa=\eta_{f}(x), $ where $f: R\rightarrow R, x\mapsto (a-1)x$ is an $R-$linear map from $R$ to $R$. So $\varphi$ is an automorphism.

When $n=2$, similar to the case $n\geq 3$, there is an automorphism $\varphi_{e_{11}}$, depending on $e_{11}$, such that $\varphi_{e_{11}}^{-1}\varphi(e_{11})=e_{11}$. Denote $\varphi_{e_{11}}^{-1}\varphi$ by $\varphi_{1}$. Clearly, $\varphi_{1}$ is also a local automorphism of $T_{2}(R)$, by Lemma 2.1, there exist an inner automorphism $\theta_{a_{2}}$ and a central automorphism $\eta_{f_{2}}$, corresponding to $e_{22}$, such that $\varphi_{1}(e_{22})=\theta_{a_{2}}\eta_{f_{2}}(e_{22})=\theta_{a_{2}}(e_{22})+f_{2}(e_{22})e.$ So

$ \begin{eqnarray} \varphi_{1}(e_{11}+e_{22}) =e_{11}+\theta_{a_{2}}(e_{22})+f_{2}(e_{22})e \equiv e+f_{2}(e_{22})e\ {\mbox{mod}}\ \textbf{n}. \end{eqnarray} $

(2.15)

On the other hand, there exist an inner automorphism $\theta_{b_{2}}$ and a central automorphism $\eta_{g_{2}}$, depending on $e_{11}+e_{22}$, such that

$ \begin{eqnarray} \varphi_{1}(e_{11}+e_{22})=\theta_{b_{2}}\eta_{g_{2}}(e_{11}+e_{22})=e+g_{2}(e)e. \end{eqnarray} $

(2.16)

From (2.15) and (2.16), we get $\theta_{a_{2}}(e_{22})=e_{22}$. Now we have $\varphi_{1}(e_{11})=e_{11}$ and $\varphi_{1}(e_{22})=e_{22}+f_{2}(e_{22})$. Let $f$ be an $R-$linear map satisfying $f(e_{11})=0, f(e_{22})=f_{2}(e_{22})$, it is easy to check that $f\in F$ and $\eta_{f}^{-1}\varphi_{1}(e_{ii})=e_{ii}\ {\mbox{for}}\ i=1, 2.$

Denote $\eta_{f}^{-1}\varphi_{1}$ by $\varphi_{2}$. Since $\varphi_{2}$ is a local automorphism, by Lemma 2.1, we have $\varphi_{2}(e_{12})=\theta_{x}(e_{12})=ae_{12}$, where $\theta_{x}$ is an inner automorphism depending on $e_{12}$ and $a\in R^{*}$. Let $z=$diag $(1, a^{-1}), $ then $\theta_{z}^{-1}\varphi_{2}(e_{ij})=e_{ij}, 1\leq i\leq j\leq 2$, which mean $\theta_{z}^{-1}\eta_{f}^{-1}\varphi_{e_{11}}^{-1}\varphi=1, $ that is $\varphi=\varphi_{e_{11}}\eta_{f}\theta_{z}$. So $\varphi$ is an automorphism.

In [14], Wang and Yu characterized the derivations of $T_{n}(R)$ by the following lemma. Before giving this lemma, we first introduce two standard derivations of $T_{n}(R)$.

(A) Inner derivations

Let $t\in T_{n}(R)$, then ad $t: x\mapsto[t, x], x\in T_{n}(R)$ is a derivation of $T_{n}(R)$, which is called an inner derivation of $T_{n}(R)$ induced by $t$.

(B) Central derivations

We denote by Hom $(D_{n}(R), R)$ the set of all $R$-module homomorphisms from $D_{n}(R)$ to $R$. For any $\sigma\in$Hom $(D_{n}(R), R)$, $\sigma$ may be extended to a derivation $\eta_{\sigma}$ of $T_{n}(R)$ by: $\eta_{\sigma}(d+x)=\sigma(d)e$ for all $d\in D_{n}(R), x\in \textbf{n}$. $\eta_{\sigma}$ is called a central derivation of $T_{n}(R)$ induced by $\sigma$.

Lemma 3.1(the theorem of [14])  Let $R$ be a commutative ring with identity. Then

(1) every derivation of $T_{n}(R)$ can be uniquely written as the sum of an inner derivation and a central derivation when $n\geq 2$.

(2) every derivation of $T_{n}(R)$ is a central derivation when $n=1$.

In order to achieve our goal, we also need other lemmas.

Lemma 3.2  Let $\delta$ be a local derivation of $T_{n}(R), n\geq 2$. If $\delta(e_{11})=0$, then there exist an inner derivation ad $m=\sum_{j=2}^{n}$ad $m_{j}$ and a central derivation $\eta_{\sigma}$ such that $(\delta-{\mbox{ad}}\ m-\eta_{\sigma})(e_{ii})=0\ {\mbox{for}}\ i=1, 2, \cdots, n.$

Proof  By the definition of $\delta$ and Lemma 3.1, there exists a derivation $\delta_{e_{22}}={\mbox{ad}}\ t_{2}+\eta_{\sigma_{2}}$, corresponding to $e_{22}$, such that

$ \begin{eqnarray} \delta(e_{11}+e_{22})=\delta(e_{11})+\delta(e_{22})=0+\delta_{e_{22}}(e_{22})={\mbox{ad}}\ t_{2}(e_{22})+\sigma_{2}(e_{22})e. \end{eqnarray} $

(3.1)

On the other hand, there is a derivation $\delta_{e_{11}+e_{22}}$=ad $s_{2}+\eta_{\alpha_{2}}$, depending on $e_{11}+e_{22}$, such that

$ \begin{eqnarray} \delta(e_{11}+e_{22})=\delta_{e_{11}+e_{22}}(e_{11}+e_{22})={\mbox{ad}}\ s_{2}(e_{11}+e_{22})+\alpha_{2}(e_{11}+e_{22})e. \end{eqnarray} $

(3.2)

Suppose $t_{2}=(t_{ij}^{(2)})_{n\times n}$, from (3.1) and (3.2), we have $t_{12}^{(2)}=0$. Let $m_{2}=(m_{ij}^{(2)})_{n\times n}$, where $m_{ij}^{(2)}=t_{ij}^{(2)}$ for $2\leq i\leq j\leq n$, and $m_{1j}^{(2)}=0$ for $1\leq j\leq n$. Then $(\delta-{\mbox{ad}}\ m_{2})(e_{11})=0$ and $(\delta-{\mbox{ad}}\ m_{2})(e_{22})=\sigma_{2}(e_{22})e$. Denote $\delta-{\mbox{ad}}\ m_{2}$ by $\delta_{1}$.

By induction we assume that there are ad $m_{j}, j=3, 4, \cdots, k-1$ such that

$ (\delta_{1}-\sum\limits_{j=3}^{k-1}{\mbox{ad}}\ m_{j})(e_{11})=0 \ {\mbox{and}}\ (\delta_{1}-\sum\limits_{j=3}^{k-1}{\mbox{ad}}\ m_{j})(e_{ii})=\sigma_{i}(e_{ii})e, $

where $\sigma_{i}\in$Hom $(D_{n}(R), R), i=2, \cdots, k-1$. Denote $\delta_{1}-\sum_{j=3}^{k-1}{\mbox{ad}}\ m_{j}$ by $\delta_{k-2}$. It is obvious that $\delta_{k-2}$ is also a local derivation. By Lemma 3.1, there exist an inner derivation ad $t_{k}$ and a central derivation $\eta_{\sigma_{k}}$, depending on $e_{kk}$, such that

$ \begin{eqnarray} &&\delta_{k-2}(e_{11}+e_{22}+\cdots+e_{kk})\nonumber\\ &=&\sigma_{2}(e_{22})e+\cdots+\sigma_{k-1}(e_{k-1, k-1})e+{\mbox{ad}}\ t_{k}(e_{kk})++\sigma_{k}(e_{kk})e. \end{eqnarray} $

(3.3)

On the other hand, since $\delta_{k-2}$ is a local derivation, we have

$ \begin{eqnarray} &&\delta_{k-2}(e_{11}+e_{22}+\cdots+e_{kk})\nonumber\\ &=&{\mbox{ad}}\ s_{k}(e_{11}+e_{22}+\cdots+e_{kk})+\alpha_{k}(e_{11}+e_{22}+\cdots+e_{kk})e, \end{eqnarray} $

(3.4)

where $s_{k}\in T_{n}(R)$ and $\alpha_{k}\in$Hom $(D_{n}(R), R)$, depending on $e_{11}+e_{22}+\cdots+e_{kk}$. Suppose $t_{k}=(t_{ij}^{(k)})_{n\times n}.$ By (3.3) and (3.4), we have $t_{jk}^{(k)}=0$ for $1\leq j\leq k-1$. Let $m_{k}=(m_{ij}^{(k)})_{n\times n}$, where $m_{ij}^{(k)}=t_{ij}^{(k)}$ for $k\leq i\leq j\leq n$, and $m_{st}^{(k)}=0$ for $1\leq s\leq k-1, s\leq t\leq n$. Then $(\delta_{k-2}-{\mbox{ad}}\ m_{k})(e_{11})=0$ and $(\delta_{k-2}-{\mbox{ad}}\ m_{k})(e_{ii})=\sigma_{i}(e_{ii})e$ for $2\leq i\leq k$.

When $k=n$, let $m=\sum\limits_{j=2}^{n}m_{j}$. Then

$ (\delta-{\mbox{ad}}\ m)(e_{11})=0, \ {\mbox{and}}\ (\delta-{\mbox{ad}}\ m)(e_{ii})= \sigma_{i}(e_{ii})e\ {\mbox{for}}\ i=2, 3, \cdots, n. $

Let $\sigma$ be an $R-$linear map from $D_{n}(R)$ to $R$, and define $\sigma(e_{11})=0, \sigma(e_{ii})=\sigma_{i}(e_{ii})$ for $i=2, 3, \cdots, n$. Then $\sigma\in$Hom $(D_{n}(R), R)$ and $(\delta-{\mbox{ad}}\ m-\eta_{\sigma})(e_{ii})=0$ for $i=1, 2, \cdots, n.$

Lemma 3.3  Let $\delta$ be a local derivation of $T_{n}(R)$ satisfying $\delta(e_{ii})=0$ for $i=1, 2, \cdots, n$. Then $\delta(e_{ij})=a_{ij}e_{ij}$ for some $a_{ij}\in R$ and $1\leq i<j\leq n$.

Proof  For $e_{ii}+e_{ij}, j\neq i$, since $\delta$ is a local derivation, from Lemma 3.1 we know there exist an inner derivation ${\mbox{ad}}\ x_{ij}$ and a central derivation $\eta_{\gamma_{ij}}$, depending on $e_{ii}+e_{ij}$, such that

$ \begin{eqnarray} \delta(e_{ii}+e_{ij})=({\mbox{ad}}\ x_{ij}+\eta_{\gamma_{ij}})(e_{ii}+e_{ij})={\mbox{ad}}\ x_{ij}(e_{ii}+e_{ij})+\gamma_{ij}(e_{ii}+e_{ij})e. \end{eqnarray} $

(3.5)

On the other hand, by the definition of $\delta$ and Lemma 3.1, we have

$ \begin{eqnarray} \delta(e_{ii}+e_{ij})=\delta(e_{ii})+\delta(e_{ij})=\delta(e_{ij})={\mbox{ad}}\ p_{ij}(e_{ij})\in \textbf{n}, \end{eqnarray} $

(3.6)

where $p_{ij}\in T_{n}(R)$ depending on $e_{ij}$. By (3.5) and (3.6), we have

$ \begin{eqnarray} {\mbox{ad}}\ x_{ij}(e_{ii}+e_{ij})={\mbox{ad}}\ p_{ij}(e_{ij}). \end{eqnarray} $

(3.7)

Similarly, there exists some $y_{ij}\in T_{n}(R)$ such that

$ \begin{eqnarray} {\mbox{ad}}\ y_{ij}(e_{jj}+e_{ij})={\mbox{ad}}\ p_{ij}(e_{ij}). \end{eqnarray} $

(3.8)

(3.7) and (3.8) imply that $\delta(e_{ij})=a_{ij}e_{ij}$ for some $a_{ij}\in R$ and $1\leq i<j\leq n$.

Lemma 3.4  Let $\delta$ be a local derivation of $T_{n}(R)$. If $\delta(e_{ii})=0$ for $i=1, 2, \cdots, n$, then there exists some $h\in T_{n}(R)$ such that $(\delta-{\mbox{ad}}\ h)(e_{i, i+1})=0 \ {\mbox{for}}\ i=1, 2, \cdots, n-1, $ and $(\delta-{\mbox{ad}}\ h)(e_{ii})=0 \ {\mbox{for}}\ i=1, 2, \cdots, n.$

Proof  By Lemma 3.3, we have $\delta(e_{i, i+1})=a_{i, i+1}e_{i, i+1}$ for some $a_{i, i+1}\in R$. Let

$ h={\mbox{diag}}(0, -a_{12}, -(a_{12}+a_{23}), \cdots, -(a_{12}+a_{23}+\cdots+a_{n-1, n})). $

Then $(\delta-{\mbox{ad}}\ h)(e_{i, i+1})=0 \ {\mbox{for}}\ i=1, 2, \cdots, n-1, $ and $(\delta-{\mbox{ad}}\ h)(e_{ii})=0 \ {\mbox{for}}\ i=1, 2, \cdots, n.$

Lemma 3.5  Let $\delta$ be a local derivation of $T_{n}(R)$ satisfying $\delta(e_{ii})=0$ for $i=1, 2, \cdots, n$, and $\delta(e_{i, i+1})=0$ for $i=1, 2, \cdots, n-1$. Then we have $\delta(e_{i, i+k})=0$ for any $e_{i, i+k}\in T_{n}(R).$

Proof  We will prove this lemma by induction on $k, k\geq 2$. When $k=2, $ for $e_{i, i+1}+e_{i+1, i+2}+e_{i, i+2}+e_{i+1, i+1}$, since $\delta$ is a local derivation, by Lemma 3.1, there exist an inner derivation ${\mbox{ad}}\ q_{i}^{(2)}$ and a central derivation $\eta_{\chi_{i}^{(2)}}$, depending on $e_{i, i+1}+e_{i+1, i+2}+e_{i, i+2}+e_{i+1, i+1}$, such that

$ \begin{eqnarray} &&\delta(e_{i, i+1}+e_{i+1, i+2}+e_{i, i+2}+e_{i+1, i+1})\nonumber\\ &=&{\mbox{ad}}\ q_{i}^{(2)}(e_{i, i+1}+e_{i+1, i+2}+e_{i, i+2}+e_{i+1, i+1})\nonumber\\ &&+ \chi_{i}^{(2)}(e_{i, i+1}+e_{i+1, i+2}+e_{i, i+2}+e_{i+1, i+1})e. \end{eqnarray} $

(3.9)

On the other hand, By Lemma 3.3, we have

$ \begin{eqnarray} \delta(e_{i, i+1}+e_{i+1, i+2}+e_{i, i+2}+e_{i+1, i+1})=a_{i, i+2}e_{i, i+2}. \end{eqnarray} $

(3.10)

From (3.9) and (3.10), we have

$ {\mbox{ad}}\ q_{i}^{(2)}(e_{i, i+1}+e_{i+1, i+2}+e_{i, i+2}+e_{i+1, i+1})=a_{i, i+2}e_{i, i+2}, $

this forces that $a_{i, i+2}=0$, that is to say $\delta(e_{i, i+2})=0$.

By induction we assume that $\delta(e_{i, i+m})=0$ for $m=2, 3, \cdots, k-1.$ For

$ e_{i, i+1}+e_{i+1, i+k}+e_{i, i+k}+e_{i+1, i+1}, $

similar to the case $k=2$, we can get there exists some $q_{i}^{(k)}\in T_{n}(R)$ such that

$ {\mbox{ad}}\ q_{i}^{(k)}(e_{i, i+1}+e_{i+1, i+k}+e_{i, i+k}+e_{i+1, i+1})=a_{i, i+k}e_{i, i+k}, $

which means that $a_{i, i+k}=0$. So $\delta(e_{i, i+k})=0$ for any $e_{i, i+k}\in T_{n}(R)$.

By those lemmas, we can prove the following theorem.

Theorem 3.1  Let $R$ be a commutative ring with identity, $T_{n}(R)$ the Lie algebra consisting of all upper triangular $n\times n$ matrices over $R$. Then every local derivation $\delta$ of $T_{n}(R)$ is a derivation.

Proof  Let $\delta$ be a local derivation of $T_{n}(R)$. When $n\geq 2$, for $e_{11}\in T_{n}(R)$, there exists a derivation $\delta_{e_{11}}$, depending on $e_{11}$, such that $\delta(e_{11})=\delta_{e_{11}}(e_{11})$. So $(\delta-\delta_{e_{11}})(e_{11})=0$. Clearly, $\delta-\delta_{e_{11}}$ is also a local derivation of $T_{n}(R)$. By Lemmas 3.2--3.5, we know there exist $\eta_{\sigma}, $ ad $m$ and ad $h$ such that

$ (\delta-{\mbox{ad}}\ m-\eta_{\sigma}-{\mbox{ad}}\ h)(e_{ij})=0\ {\mbox{for}}\ 1\leq i\leq j\leq n, $

which imply that $\delta={\mbox{ad}}\ m+\eta_{\sigma}+{\mbox{ad}}\ h$, so $\delta$ is a derivation.

When $n=1$, suppose that $\delta(1)=b$, then for any $x\in T_{1}(R)=R$, we have

$ \delta(x)=x\delta(1)=xb=\eta_{\sigma}(x), $

where $\sigma: R\rightarrow R, x\mapsto bx$ is an $R$-linear from $R$ to $R$. So $\delta$ is a derivation.