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 $$\bigoplus_\Delta R_\alpha$$ is a right ideal of $$\prod_\Delta R_\alpha$$ Proof ... Let $$(x_\alpha ) , (y_\alpha ) \in \bigoplus_\Delta R_\alpha$$ and let $$(r_\alpha ) \in \prod_\Delta R_\alpha$$Then $$ (x_\alpha ) + (y_\alpha ) = (x_\alpha + y_\alpha )$$ ... by the rule of addition in direct products ...Now ... $$x_\alpha + y_\alpha \in R_\alpha$$ for all $$\alpha \in \Delta$$ ... by closure of addition in rings ... Thus $$(x_\alpha + y_\alpha ) \in \prod_\Delta R_\alpha$$ ...... but also ... since $$(x_\alpha $$) and $$(y_\alpha )$$ each have only a finite number of non-zero components ...

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

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

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

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

and ...

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

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

$$\Longrightarrow (x_\alpha) (r_\alpha) \in \bigoplus_\Delta R_\alpha$$ ... ... ... ... ... (2)
$$(1) (2) \Longrightarrow$$ $$\bigoplus_\Delta R_\alpha$$ is a right ideal of $$\prod_\Delta R_\alpha$$
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 $\prod_\Delta R_\alpha$