I am reading An Introduction to Rings and Modules With K-Theory in View by A.J. Berrick and M.E. Keating (B&K).(adsbygoogle = window.adsbygoogle || []).push({});

In Chapter2: Direct Sums and Short Exact Sequences in Section 2.1.16 B&K deal with the standard free right ##R##-module on a set ##X##. I need some help with understanding an aspect of the authors' discussion ... ...

Section 2.1.16 reads as follows:

In the above text by B&K they construct the standard free module on ##X## as the module

##\text{Fr}_R (x) = \bigoplus_X xR##

so that the elements of ##\text{Fr}_R (x)## are formal sums ##m = \sum_{x \in X}x r_x (m)##

with ##r_x (m) \in R## ... ...

OK ... so far so good ... BUT ...

... can someone please explain to me how this enables B&K to say ...

" ... ... It is clear that if ##\Lambda## is any convenient ordering of ##X##, we have

##\text{Fr}_R (x) = R^{ \Lambda }## ... ... "

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To make my question more explicit ... suppose that ##X## is a finite ordered set ##X = \{ x_1, x_2, x_3 \}##

Then ##\text{Fr}_R (x) = \bigoplus_X xR = x_1 R + x_2 R + x_3 R##

... ... and elements of ##\text{Fr}_R (x)## would be of the form ##m = x_1 r_1 + x_2 r_2 + x_3 r_3##

BUT ... to repeat my question ... with elements of this form how can we argue that

if ##\Lambda## is any convenient ordering of ##X##, we have ##\text{Fr}_R (x) = R^{ \Lambda }## ... ?

Note: I suspect that B&K may be expecting the reader to identify ##\sum x r_x(m)## with ##\sum r_x(m)## ....

... BUT ... if this is the case ... why introduce ##X## into a construction only to "identify" it away ... why not just define the standard free right ##R##-module as ##\text{Fr}_R (x) = R^{ \Lambda }## ?

Hope someone can help ...

Peter

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# I Standard Free Right R-Module on X - B&K Section 2.1.16

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