- #1
Math Amateur
Gold Member
MHB
- 3,998
- 48
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(b) of Problem Set 2.1 ...
Problem 2(b) 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 A_\alpha## is a right ideal of ##\prod_\Delta R_\alpha##Proof ...Let ##(x_\alpha ) , (y_\alpha ) \in \bigoplus_\Delta A_\alpha## and let ##(r_\alpha ) \in \prod_\Delta R_\alpha##Then ## (x_\alpha ) + (y_\alpha ) = (x_\alpha + y_\alpha )## ...... further ... if ##(x_\alpha )## has ##m## non-zero components and ##(y_\alpha )## has ##n## non-zero components then ##(x_\alpha + y_\alpha )## has at most ##(m+n)## non-zero components ... that is ##(x_\alpha + y_\alpha )## has only a finite number of non-zero components ...
... so ... since each ##x_\alpha + y_\alpha \in A_\alpha## we have that ##(x_\alpha + y_\alpha ) \in \bigoplus_\Delta A_\alpha## ...
Hence ... ##(x_\alpha ) + (y_\alpha ) \in \bigoplus_\Delta A_\alpha## ... ... ... ... ... (1)
Also ... we have ...
##(x_\alpha ) (r_\alpha ) = (x_\alpha r_\alpha)##
... and assuming ##x_\alpha## has ##m## non-zero components, then ##(x_\alpha r_\alpha)## has at most ##m## non-zero components ...... and ...##x_\alpha r_\alpha \in A_\alpha## since ##A_\alpha## is a right ideal of ##R_\alpha##so ##(x_\alpha r_\alpha) \in \bigoplus_\Delta A_\alpha##and it follows that ## (x_\alpha) ( r_\alpha) \in \bigoplus_\Delta A_\alpha## ... ... ... ... ... (2)##(1) (2) \Longrightarrow \bigoplus_\Delta A_\alpha## is a right ideal of ##\prod_\Delta R_\alpha##
Can someone please critique my proof either by confirming it to be correct and/or pointing out errors and shortcomings ...
Peter