How Do You Interpret Rowen's Notation in Matrix Rings?

[Math Amateur]

I am reading Louis Rowen's book, "Ring Theory"(Student Edition) ...

I have a problem interpreting Rowen's notation in Section 1.1 Matrix Rings and Idempotents ...

The relevant section of Rowen's text reads as follows:View attachment 6069
View attachment 6070In the above text from Rowen, we read the following:
" ... ... We obtain a more explicit notation by defining the \(\displaystyle n \times n\) matric unit \(\displaystyle e_{ij}\) to be the matrix whose \(\displaystyle i-j\) entry is \(\displaystyle 1\), with all other entries \(\displaystyle 0\).Thus \(\displaystyle ( r_{ij} ) = \sum_{i,j =1}^n r_{ij} e_{ij}\) ; addition is componentwise and multiplication is given according to the rule \(\displaystyle ( r_1 e_{ij} ) ( r_2 e_{uv} ) = \delta_{ju} (r_1 r_2) e_{iv} \)

... ... ...
I am having trouble understanding the rule \(\displaystyle ( r_1 e_{ij} ) ( r_2 e_{uv} ) = \delta_{ju} (r_1 r_2) e_{iv}\) ... ...

What are \(\displaystyle r_1\) and \(\displaystyle r_2\) ... where exactly do they come from ... ?

Can someone please explain the rule to me ...?To take a specific example ... suppose we are dealing with \(\displaystyle M_2 ( \mathbb{Z} )\) and we have two matrices ...\(\displaystyle P = \begin{pmatrix} 1 & 3 \\ 5 & 4 \end{pmatrix}\)

and

\(\displaystyle Q = \begin{pmatrix} 2 & 1 \\ 3 & 3 \end{pmatrix}\) ...In this specific case, what are \(\displaystyle r_1\) and \(\displaystyle r_2\) ... ... and how would the rule in question work ...?
Hope someone can help ...

Peter

 

[Evgeny.Makarov]

I am having trouble understanding the rule \(\displaystyle ( r_1 e_{ij} ) ( r_2 e_{uv} ) = \delta_{ju} (r_1 r_2) e_{iv}\) ... ...

What are \(\displaystyle r_1\) and \(\displaystyle r_2\) ... where exactly do they come from ... ?

$r_1$ and $r_2$ are numbers, or, more precisely, elements of the ring $R$. They are also coefficients, or coordinates, of a matrix in the basis consisting of $e_{ij}$. If one matrix is $\displaystyle\sum_{i,j=1}^nr^{(1)}_{ij}e_{ij}$ and another is $\displaystyle\sum_{i,j=1}^nr^{(2)}_{ij}e_{ij}$, then when you multiply them, you apply distributivity and get $\displaystyle\sum_{i,j,u,v=1}^nr^{(1)}_{ij}e_{ij}r^{(2)}_{uv}e_{uv}$, and the formula in your quote says how to compute $r^{(1)}_{ij}e_{ij}r^{(2)}_{uv}e_{uv}$.

 

[Math Amateur]

$r_1$ and $r_2$ are numbers, or, more precisely, elements of the ring $R$. They are also coefficients, or coordinates, of a matrix in the basis consisting of $e_{ij}$. If one matrix is $\displaystyle\sum_{i,j=1}^nr^{(1)}_{ij}e_{ij}$ and another is $\displaystyle\sum_{i,j=1}^nr^{(2)}_{ij}e_{ij}$, then when you multiply them, you apply distributivity and get $\displaystyle\sum_{i,j,u,v=1}^nr^{(1)}_{ij}e_{ij}r^{(2)}_{uv}e_{uv}$, and the formula in your quote says how to compute $r^{(1)}_{ij}e_{ij}r^{(2)}_{uv}e_{uv}$.

Thanks Evgeny ... your post was very clear ... and most helpfu ...

Peter