- #1
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Homework Statement
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:
Homework Equations
The Attempt at a Solution
My 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