- #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 help with Problem 2(a) of Problem Set 2.1 ...
Problem 2(a) of Problem Set 2.1 reads as follows:
I 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 ...
Homework Equations
The definition of a right ideal is relevant to this problem ... Bland's definition of a right ideal is as follows:
The Attempt at a Solution
My attempted solution to problem 2(a) is as follows:... we have to show that ##\prod_\Delta A_\alpha## is a right ideal of ##\prod_\Delta R_\alpha## ...To demonstrate this we have to show that ##\prod_\Delta A_\alpha## is closed under addition and closed under multiplication on the right by an element of ##\prod_\Delta R_\alpha## ...So ... let ##(x_\alpha), (y_\alpha) \in \prod_\Delta A_\alpha## and ##(r_\alpha) \in \prod_\Delta R_\alpha##
Then ##x_\alpha, y_\alpha \in A_\alpha## for all ##\alpha \in \Delta##
##\Longrightarrow x_\alpha + y_\alpha \in A_\alpha## since ##A_\alpha## is a right ideal of ##R_\alpha## for all ##\alpha \in \Delta## ...
##\Longrightarrow (x_\alpha) + (y_\alpha) \in \prod_\Delta A_\alpha##
##\Longrightarrow \prod_\Delta A_\alpha## is closed under addition ...
Now ... ##(x_\alpha) \in \prod_\Delta A_\alpha , (r_\alpha) \in \prod_\Delta R_\alpha##
##\Longrightarrow x_\alpha \in A_\alpha , r_\alpha \in R_\alpha## for all ##\alpha \in \Delta## ...
##\Longrightarrow x_\alpha r_\alpha \in A_\alpha## since ##A_\alpha## is a right ideal of ##R_\alpha## ...
##\Longrightarrow ( x_\alpha r_\alpha ) \in \prod_\Delta A_\alpha##Thus ##\prod_\Delta A_\alpha## is a right ideal of ##\prod_\Delta R_\alpha## ...
Hope the above is correct ...
Peter