I am reading Dummit and Foote's book: Abstract Algebra ... ... and am currently focused on Section 10.4 Tensor Products of Modules ... ...(adsbygoogle = window.adsbygoogle || []).push({});

I have a basic question regarding the extension of the scalars ...

Dummit and Foote's (D&Fs) exposition regarding extension of the scalars reads as follows:

Question 1

In the above text from D&F (towards the end of the quote) we read the following:

"... ... Suppose now that ##\sum s_i \otimes n_i = \sum s'_i \otimes n'_i##

are two representations for the same element in ##S \otimes_R N##. Then ##\sum (s_i, n_i) - \sum (s'_i, n'_i)## is an element of ##H## ... ... ... "

Can someone please explain exactly why ##\sum s_i \otimes n_i = \sum s'_i \otimes n'_i## in ##S \otimes_R N## implies that ##\sum (s_i, n_i) - \sum (s'_i, n'_i)## is an element of ##H## ... ...

[ ***Note*** I am a little unsure of the general nature of elements of ##H## ... and even more unsure of the nature of elements of ##S \otimes_R N## ... ... ]

Question 2

In the above text from D&F (towards the end of the quote) we read the following:

"... ... for any ##s \in S## also ##\sum (ss_i, n_i) - \sum (ss'_i, n'_i)## is an element of ##H##. But this means that ##\sum ss_i \otimes n_i - \sum ss'_i \otimes n'_i)## in ##S \otimes_R N## ... ... "

Can someone please explain exactly why if ##\sum (ss_i, n_i) - \sum (ss'_i, n'_i)## is an element of ##H## ... ... that we then have ##\sum ss_i \otimes n_i - \sum ss'_i \otimes n'_i)## in ##S \otimes_R N## ... ...

... ... although the above seems right, why exactly is it the case ... ?

Hope someone can help ... I suspect my main problem is the general nature and characteristics of elements of ##H## and elements of ##S \otimes_R N##

Peter

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# I Extension of scalars ... D&F, Section 10.4: Tensor Products

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