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I am reading Dummit and Foote's book: Abstract Algebra ... ... and am currently focused on Section 10.4 Tensor Products of Modules ... ...
I have a basic question regarding the extension of the scalars ...
Dummit and Foote's exposition regarding extension of the scalars reads as follows:https://www.physicsforums.com/attachments/5491
View attachment 5492
View attachment 5493
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
I have a basic question regarding the extension of the scalars ...
Dummit and Foote's exposition regarding extension of the scalars reads as follows:https://www.physicsforums.com/attachments/5491
View attachment 5492
View attachment 5493
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