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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 $$n \times n$$ matric unit $$e_{ij}$$ to be the matrix whose $$i-j$$ entry is $$1$$, with all other entries $$0$$.Thus $$( r_{ij} ) = \sum_{i,j =1}^n r_{ij} e_{ij}$$ ; addition is componentwise and multiplication is given according to the rule $$( r_1 e_{ij} ) ( r_2 e_{uv} ) = \delta_{ju} (r_1 r_2) e_{iv} $$
... ... ...
I am having trouble understanding the rule $$( r_1 e_{ij} ) ( r_2 e_{uv} ) = \delta_{ju} (r_1 r_2) e_{iv}$$ ... ...
What are $$r_1$$ and $$r_2$$ ... where exactly do they come from ... ?
Can someone please explain the rule to me ...?To take a specific example ... suppose we are dealing with $$M_2 ( \mathbb{Z} )$$ and we have two matrices ...$$P = \begin{pmatrix} 1 & 3 \\ 5 & 4 \end{pmatrix}$$
and
$$Q = \begin{pmatrix} 2 & 1 \\ 3 & 3 \end{pmatrix}$$ ...In this specific case, what are $$r_1$$ and $$r_2$$ ... ... and how would the rule in question work ...?
Hope someone can help ...
Peter
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 $$n \times n$$ matric unit $$e_{ij}$$ to be the matrix whose $$i-j$$ entry is $$1$$, with all other entries $$0$$.Thus $$( r_{ij} ) = \sum_{i,j =1}^n r_{ij} e_{ij}$$ ; addition is componentwise and multiplication is given according to the rule $$( r_1 e_{ij} ) ( r_2 e_{uv} ) = \delta_{ju} (r_1 r_2) e_{iv} $$
... ... ...
I am having trouble understanding the rule $$( r_1 e_{ij} ) ( r_2 e_{uv} ) = \delta_{ju} (r_1 r_2) e_{iv}$$ ... ...
What are $$r_1$$ and $$r_2$$ ... where exactly do they come from ... ?
Can someone please explain the rule to me ...?To take a specific example ... suppose we are dealing with $$M_2 ( \mathbb{Z} )$$ and we have two matrices ...$$P = \begin{pmatrix} 1 & 3 \\ 5 & 4 \end{pmatrix}$$
and
$$Q = \begin{pmatrix} 2 & 1 \\ 3 & 3 \end{pmatrix}$$ ...In this specific case, what are $$r_1$$ and $$r_2$$ ... ... and how would the rule in question work ...?
Hope someone can help ...
Peter