Checking My Solution to Problem 2(c) of Problem Set 2.1: Seeking Critique

[Math Amateur]

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 someone to check my solution to Problem 2(c) of Problem Set 2.1 ...

Problem 2(c) of Problem Set 2.1 reads as follows:View attachment 8061My attempt at a solution follows:We claim that \(\displaystyle \bigoplus_\Delta R_\alpha\) is a right ideal of \(\displaystyle \prod_\Delta R_\alpha\) Proof ... Let \(\displaystyle (x_\alpha ) , (y_\alpha ) \in \bigoplus_\Delta R_\alpha\) and let \(\displaystyle (r_\alpha ) \in \prod_\Delta R_\alpha\)Then \(\displaystyle (x_\alpha ) + (y_\alpha ) = (x_\alpha + y_\alpha )\) ... by the rule of addition in direct products ...Now ... \(\displaystyle x_\alpha + y_\alpha \in R_\alpha\) for all \(\displaystyle \alpha \in \Delta\) ... by closure of addition in rings ... Thus \(\displaystyle (x_\alpha + y_\alpha ) \in \prod_\Delta R_\alpha\) ...... but also ... since \(\displaystyle (x_\alpha \)) and \(\displaystyle (y_\alpha )\) each have only a finite number of non-zero components ...

... we have that \(\displaystyle (x_\alpha + y_\alpha )\) has only a finite number of non-zero components ...

... so ... \(\displaystyle (x_\alpha + y_\alpha ) \in \bigoplus_\Delta R_\alpha\) ...

Hence \(\displaystyle (x_\alpha ) + (y_\alpha ) \in \bigoplus_\Delta R_\alpha \) ... ... ... ... ... (1)
Now we also have that ... \(\displaystyle (x_\alpha ) (r_\alpha ) = (x_\alpha r_\alpha)\) ... ... rule of multiplication in a direct product ...

Now ... \(\displaystyle x_\alpha r_\alpha \in R_\alpha\) for all \(\displaystyle \alpha \in \Delta\) ... since a ring is closed under multiplication ...

and ...

\(\displaystyle (x_\alpha r_\alpha)\) has only a finite number of non-zero components since \(\displaystyle (x_\alpha )\) has only a finite number of non-zero components ...

So ... \(\displaystyle (x_\alpha r_\alpha) \in \bigoplus_\Delta R_\alpha\)

\(\displaystyle \Longrightarrow (x_\alpha) (r_\alpha) \in \bigoplus_\Delta R_\alpha\) ... ... ... ... ... (2)
\(\displaystyle (1) (2) \Longrightarrow\) \(\displaystyle \bigoplus_\Delta R_\alpha\) is a right ideal of \(\displaystyle \prod_\Delta R_\alpha\)
Can someone please critique my proof ... ... and either confirm its correctness or point out the errors and shortcomings ...

Such help will be much appreciated ...

Peter

 

[steenis]

This is correct, Peter.
You can also say: $R_\alpha$ is a right ideal of $R_\alpha$, then apply (b).
But you forgot something.

(c) asked is $\bigoplus_\Delta R_\alpha$ an ideal of $\prod_\Delta R_\alpha$ ?

So you have to prove that $\bigoplus_\Delta R_\alpha$ a two-sided ideal of $\prod_\Delta R_\alpha$