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I am reading Paul E. Bland's book "Rings and Their Modules ...
Currently I am focused on Section 2.3 Tensor Products of Modules ... ...
I need some help in order to fully understand the Remark that Bland makes on Pages 65- 66
Bland's remark reads as follows:View attachment 5648
View attachment 5651
Question 1
In the above text by Bland we read the following:
"... ... but when \(\displaystyle g\) is specified in this manner it is difficult to show that it is well defined ... ... "
What does Bland mean by showing \(\displaystyle g\) is well defined and why would this be difficult ... ...Reflection on Question 1Reflecting ... ... I suspect that when Bland talks about \(\displaystyle g\) being "well defined" he means that if we choose a different element ... say, \(\displaystyle \sum_{ i = 1}^m n'_i ( x'_i \otimes y'_i )\) in the same coset as \(\displaystyle \sum_{ i = 1}^m n_i ( x_i \otimes y_i )\) ... ... then \(\displaystyle g\) still maps onto \(\displaystyle \sum_{ i = 1}^m n_i ( f(x_i) \otimes y_i ) \) ... ... is that correct ...
Question 2
In the above text by Bland we read the following:
"... ... Since the map \(\displaystyle h = \rho' ( f \times id_N )\) is an R-balanced map ... ... "Why is \(\displaystyle h = \rho' ( f \times id_N )\) an R-balanced map ... can someone please demonstrate that this is the case?
Hope someone can help ... ...
Peter==================================================================================The following text including some relevant definitions may be useful to readers not familiar with Bland's textbook... note in particular the R-module in Bland's text means right R-module ...
View attachment 5650
Currently I am focused on Section 2.3 Tensor Products of Modules ... ...
I need some help in order to fully understand the Remark that Bland makes on Pages 65- 66
Bland's remark reads as follows:View attachment 5648
View attachment 5651
Question 1
In the above text by Bland we read the following:
"... ... but when \(\displaystyle g\) is specified in this manner it is difficult to show that it is well defined ... ... "
What does Bland mean by showing \(\displaystyle g\) is well defined and why would this be difficult ... ...Reflection on Question 1Reflecting ... ... I suspect that when Bland talks about \(\displaystyle g\) being "well defined" he means that if we choose a different element ... say, \(\displaystyle \sum_{ i = 1}^m n'_i ( x'_i \otimes y'_i )\) in the same coset as \(\displaystyle \sum_{ i = 1}^m n_i ( x_i \otimes y_i )\) ... ... then \(\displaystyle g\) still maps onto \(\displaystyle \sum_{ i = 1}^m n_i ( f(x_i) \otimes y_i ) \) ... ... is that correct ...
Question 2
In the above text by Bland we read the following:
"... ... Since the map \(\displaystyle h = \rho' ( f \times id_N )\) is an R-balanced map ... ... "Why is \(\displaystyle h = \rho' ( f \times id_N )\) an R-balanced map ... can someone please demonstrate that this is the case?
Hope someone can help ... ...
Peter==================================================================================The following text including some relevant definitions may be useful to readers not familiar with Bland's textbook... note in particular the R-module in Bland's text means right R-module ...
View attachment 5650
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