I have a doubt in Dummit's Abstract Algebra on page245 :
"For each $j\in[1,2,\dots,n]$ let $L_j$ be the set of all $n\times n$ matrices in $M_n(R)$ with arbitrary entries in the $j^{\text{th}}$ column and zeros in all other columns."
By the argument above, any matrix in $L_j$ must be of the form as follows: $$ A=\begin{bmatrix} \ 0&\cdots&0&a_{1j}&0&\cdots&0\ \\\ 0&\cdots&0&a_{2j}&0&\cdots&0\ \\\ \vdots&&\vdots&\vdots&\vdots&&\vdots\ \\\ 0&\cdots&0&a_{(n-1)j}&0&\cdots&0\ \\\ 0&\cdots&0&a_{nj}&0&\cdots&0\ \end{bmatrix} $$ where $a_{ij}\in R$ is arbitrary.
$\cdots\cdots$
"Moreover, $L_j$ is not a right ideal (hence is not a two-sided ideal). To see this, let $E_{pq}$ be the matrix with $1$ in the $p^{\text{th}}$ row and $q^{\text{th}}$ column and zeros elsewhere ( $p,q\in\{1,\dots,n\}$ ). Then ${\color{red}{E_{1j}\in L_j}}$ but $E_{1j}E_{ji}=E_{1i}\not\in L_j$ if $i\ne j$ , so $L_j$ is not closed under right multiplication by arbitrary ring elements."
It is clear to see that $$ E_{1j}=\begin{bmatrix} \ 1&\cdots&1&1&1&\cdots&1\ \\\ 0&\cdots&0&1&0&\cdots&0\ \\\ \vdots&&\vdots&\vdots&\vdots&&\vdots\ \\\ 0&\cdots&0&1&0&\cdots&0\ \\\ 0&\cdots&0&1&0&\cdots&0\ \end{bmatrix} $$ is not in $L_j$ , which contradicts the red part in the context.
So is it a typo? And how to show that $L_j$ is not a right ideal by correcting the statement above?
That's poor (imprecise) wording rather than a typo. A matrix unit $E_{pq}$ is a matrix that has all entries zero, except the single entry at the intersection (not union) of the $p$-th row and $q$-th column. This might indeed be confusing if you're for example self-studying and this is your first encounter with $E_{ij}$, especially since the book doesn't introduce the standard term for it right away.