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I am reading Paul E. Bland's book: Rings and Their Modules and am currently focused on Section 2.1 Direct Products and Direct Sums ... ...
I need help with Problem 2(a) of Problem Set 2.1 ...
Problem 2(a) of Problem Set 2.1 reads as follows:
View attachment 8049I am unsure of my solution to problem 2(a) and need help in the following way ...
... could someone please confirm my solution is correct and/or point out errors and shortcomings ...
... indeed I would be grateful if someone could critique my solution ...
My attempted solution to problem 2(a) is as follows:... we have to show that \(\displaystyle \prod_\Delta A_\alpha\) is a right ideal of \(\displaystyle \prod_\Delta R_\alpha\) ...
To demonstrate this we have to show that \(\displaystyle \prod_\Delta A_\alpha\) is closed under addition and closed under multiplication on the right by an element of \(\displaystyle \prod_\Delta R_\alpha\) ...So ... let \(\displaystyle (x_\alpha), (y_\alpha) \in \prod_\Delta A_\alpha\) and \(\displaystyle (r_\alpha) \in \prod_\Delta R_\alpha\)
Then \(\displaystyle x_\alpha, y_\alpha \in A_\alpha\) for all \(\displaystyle \alpha \in \Delta\)
\(\displaystyle \Longrightarrow x_\alpha + y_\alpha \in A_\alpha\) since \(\displaystyle A_\alpha\) is a right ideal of \(\displaystyle R_\alpha\) for all \(\displaystyle \alpha \in \Delta\) ...
\(\displaystyle \Longrightarrow (x_\alpha) + (y_\alpha) \in \prod_\Delta A_\alpha\)
\(\displaystyle \Longrightarrow \prod_\Delta A_\alpha\) is closed under addition ...
Now ... \(\displaystyle (x_\alpha) \in \prod_\Delta A_\alpha , (r_\alpha) \in \prod_\Delta R_\alpha\)
\(\displaystyle \Longrightarrow x_\alpha \in A_\alpha , r_\alpha \in R_\alpha\) for all \(\displaystyle \alpha \in \Delta\) ...
\(\displaystyle \Longrightarrow x_\alpha r_\alpha \in A_\alpha\) since \(\displaystyle A_\alpha\) is a right ideal of \(\displaystyle R_\alpha\) ...
\(\displaystyle \Longrightarrow ( x_\alpha r_\alpha ) \in \prod_\Delta A_\alpha\)Thus \(\displaystyle \prod_\Delta A_\alpha\) is a right ideal of \(\displaystyle \prod_\Delta R_\alpha\) ...
Hope the above is correct ...
Peter
I need help with Problem 2(a) of Problem Set 2.1 ...
Problem 2(a) of Problem Set 2.1 reads as follows:
View attachment 8049I am unsure of my solution to problem 2(a) and need help in the following way ...
... could someone please confirm my solution is correct and/or point out errors and shortcomings ...
... indeed I would be grateful if someone could critique my solution ...
My attempted solution to problem 2(a) is as follows:... we have to show that \(\displaystyle \prod_\Delta A_\alpha\) is a right ideal of \(\displaystyle \prod_\Delta R_\alpha\) ...
To demonstrate this we have to show that \(\displaystyle \prod_\Delta A_\alpha\) is closed under addition and closed under multiplication on the right by an element of \(\displaystyle \prod_\Delta R_\alpha\) ...So ... let \(\displaystyle (x_\alpha), (y_\alpha) \in \prod_\Delta A_\alpha\) and \(\displaystyle (r_\alpha) \in \prod_\Delta R_\alpha\)
Then \(\displaystyle x_\alpha, y_\alpha \in A_\alpha\) for all \(\displaystyle \alpha \in \Delta\)
\(\displaystyle \Longrightarrow x_\alpha + y_\alpha \in A_\alpha\) since \(\displaystyle A_\alpha\) is a right ideal of \(\displaystyle R_\alpha\) for all \(\displaystyle \alpha \in \Delta\) ...
\(\displaystyle \Longrightarrow (x_\alpha) + (y_\alpha) \in \prod_\Delta A_\alpha\)
\(\displaystyle \Longrightarrow \prod_\Delta A_\alpha\) is closed under addition ...
Now ... \(\displaystyle (x_\alpha) \in \prod_\Delta A_\alpha , (r_\alpha) \in \prod_\Delta R_\alpha\)
\(\displaystyle \Longrightarrow x_\alpha \in A_\alpha , r_\alpha \in R_\alpha\) for all \(\displaystyle \alpha \in \Delta\) ...
\(\displaystyle \Longrightarrow x_\alpha r_\alpha \in A_\alpha\) since \(\displaystyle A_\alpha\) is a right ideal of \(\displaystyle R_\alpha\) ...
\(\displaystyle \Longrightarrow ( x_\alpha r_\alpha ) \in \prod_\Delta A_\alpha\)Thus \(\displaystyle \prod_\Delta A_\alpha\) is a right ideal of \(\displaystyle \prod_\Delta R_\alpha\) ...
Hope the above is correct ...
Peter