- #1

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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