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matrix unit
A matrix unit is a matrix (over some ring with $1$) whose entries are all $0$ except in one cell, where it is $1$.
For example, among the $3\times 2$ matrices,
$\begin{pmatrix}1&0\\ 0&0\\ 0&0\end{pmatrix},\quad\begin{pmatrix}0&1\\ 0&0\\ 0&0\end{pmatrix},\quad\begin{pmatrix}0&0\\ 1&0\\ 0&0\end{pmatrix},\quad\begin{pmatrix}0&0\\ 0&1\\ 0&0\end{pmatrix},\quad\begin{pmatrix}0&0\\ 0&0\\ 1&0\end{pmatrix},\quad\begin{pmatrix}0&0\\ 0&0\\ 0&1\end{pmatrix}$ 
are the matrix units.
Let $A$ and $B$ be $m\times n$ and $p\times q$ matrices over $R$, and $U_{{ij}}$ an $n\times p$ matrix unit (over $R$). Then
Remarks. Let $M=M_{{m\times n}}(R)$ be the set of all $m$ by $n$ matrices with entries in a ring $R$ (with $1$). Denote $U_{{ij}}$ the matrix unit in $M$ whose cell $(i,j)$ is $1$.

$M$ is a (left or right) $R$module generated by the $m\times n$ matrix units.

When $m=n$, $M$ has the structure of an algebra over $R$. The matrix units have the following properties:
(a) $U_{{ij}}U_{{k\ell}}=\delta_{{jk}}U_{{i\ell}}$, and
(b) $U_{{11}}+\cdots+U_{{nn}}=I_{n}$,
where $\delta_{{ij}}$ is the Kronecker delta and $I_{n}$ is the identity matrix. Note that the $U_{{ii}}$ form a complete set of pairwise orthogonal idempotents, meaning $U_{{ii}}U_{{ii}}=U_{{ii}}$ and $U_{{ii}}U_{{jj}}=0$ if $i\neq j$.

In general, in a matrix ring $S$ (consisting of, say, all $n\times n$ matrices), any set of $n$ matrices satisfying the two properties above is called a full set of matrix units of $S$.

For example, if $\{U_{{ij}}\mid 1\leq i,j\leq 2\}$ is the set of $2\times 2$ matrix units over $\mathbb{R}$, then for any invertible matrix $T$, $\{TU_{{ij}}T^{{1}}\mid 1\leq i,j\leq 2\}$ is a full set of matrix units.

If we embed $R$ as a subring of $M_{n}(R)$, then $R$ is the centralizer of the matrix units of $M_{n}(R)$, meaning that the only elements in $M_{n}(R)$ that commute with the matrix units are the elements in $R$.
References
 1 T. Y. Lam, Lectures on Modules and Rings, Springer, New York, 1998.
Mathematics Subject Classification
15A30 no label found16S50 no label found Forums
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