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I am reading Paul E. Bland's book: Rings and Their Modules and am currently focused on Section 2.1 Direct Products and Direct Sums ... ...
I need help with some aspects of the proof of Proposition 2.1.1 ...
Proposition 2.1.1 and its proof read as follows:
View attachment 8030
In the statement of the above proposition we read the following:
" ... ... for every $$R$$-module $$N$$ and every family $$\{ f_\alpha \ : \ N \rightarrow M_\alpha \}_\Delta$$ of $$R$$-linear mappings there is a unique $$R$$-linear mapping $$f \ : \ N \rightarrow \prod_\Delta M_\alpha$$ ... ... "The proposition declares the family of mappings $$\{ f_\alpha \ : \ N \rightarrow M_\alpha \}_\Delta$$ as $$R$$-linear mappings and also declares that $$f$$ (see below for definition of $$f$$!) is an $$R$$-linear mapping ...
... BUT ...
I cannot see where in the proof the fact that they are $$R$$-linear mappings is used ...
Can someone please explain where in the proof the fact that the family of mappings $$\{ f_\alpha \ : \ N \rightarrow M_\alpha \}_\Delta$$ and $$f$$ are $$R$$-linear mappings is used ... basically ... why do these mappings have to be $$R$$-linear ... ?
Help will be much appreciated ...
Peter======================================================================================The above post mentions but does not define $$f$$ ... Bland's definition of $$f$$ is as follows:
View attachment 8031Hope that helps ...
Peter***EDIT***
In respect of $$f$$ it seems we have to prove $$f$$ is an $$R$$-linear mapping ... but then ... where is this done ...
I need help with some aspects of the proof of Proposition 2.1.1 ...
Proposition 2.1.1 and its proof read as follows:
View attachment 8030
In the statement of the above proposition we read the following:
" ... ... for every $$R$$-module $$N$$ and every family $$\{ f_\alpha \ : \ N \rightarrow M_\alpha \}_\Delta$$ of $$R$$-linear mappings there is a unique $$R$$-linear mapping $$f \ : \ N \rightarrow \prod_\Delta M_\alpha$$ ... ... "The proposition declares the family of mappings $$\{ f_\alpha \ : \ N \rightarrow M_\alpha \}_\Delta$$ as $$R$$-linear mappings and also declares that $$f$$ (see below for definition of $$f$$!) is an $$R$$-linear mapping ...
... BUT ...
I cannot see where in the proof the fact that they are $$R$$-linear mappings is used ...
Can someone please explain where in the proof the fact that the family of mappings $$\{ f_\alpha \ : \ N \rightarrow M_\alpha \}_\Delta$$ and $$f$$ are $$R$$-linear mappings is used ... basically ... why do these mappings have to be $$R$$-linear ... ?
Help will be much appreciated ...
Peter======================================================================================The above post mentions but does not define $$f$$ ... Bland's definition of $$f$$ is as follows:
View attachment 8031Hope that helps ...
Peter***EDIT***
In respect of $$f$$ it seems we have to prove $$f$$ is an $$R$$-linear mapping ... but then ... where is this done ...
Last edited: