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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 $$g$$ is specified in this manner it is difficult to show that it is well defined ... ... "
What does Bland mean by showing $$g$$ is well defined and why would this be difficult ... ...Reflection on Question 1Reflecting ... ... I suspect that when Bland talks about $$g$$ being "well defined" he means that if we choose a different element ... say, $$\sum_{ i = 1}^m n'_i ( x'_i \otimes y'_i )$$ in the same coset as $$\sum_{ i = 1}^m n_i ( x_i \otimes y_i )$$ ... ... then $$g$$ still maps onto $$\sum_{ i = 1}^m n_i ( f(x_i) \otimes y_i ) $$ ... ... is that correct ...
Question 2
In the above text by Bland we read the following:
"... ... Since the map $$h = \rho' ( f \times id_N )$$ is an R-balanced map ... ... "Why is $$h = \rho' ( f \times id_N )$$ 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 $$g$$ is specified in this manner it is difficult to show that it is well defined ... ... "
What does Bland mean by showing $$g$$ is well defined and why would this be difficult ... ...Reflection on Question 1Reflecting ... ... I suspect that when Bland talks about $$g$$ being "well defined" he means that if we choose a different element ... say, $$\sum_{ i = 1}^m n'_i ( x'_i \otimes y'_i )$$ in the same coset as $$\sum_{ i = 1}^m n_i ( x_i \otimes y_i )$$ ... ... then $$g$$ still maps onto $$\sum_{ i = 1}^m n_i ( f(x_i) \otimes y_i ) $$ ... ... is that correct ...
Question 2
In the above text by Bland we read the following:
"... ... Since the map $$h = \rho' ( f \times id_N )$$ is an R-balanced map ... ... "Why is $$h = \rho' ( f \times id_N )$$ 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
Last edited: