Direct Products of Modules .... Bland Proposition 2.1.1 ....

[Math Amateur]

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:

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 (that is, homomorphisms...) ... ?
Help will be much appreciated ...

Peter======================================================================================The above post mentions but does not define ##f## ... Bland's definition of ##f## is as follows:

Hope 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 ...

 

[fresh_42]

I cannot see where in the proof the fact that they are ##R-##linear mappings is used ...

Have you tried to prove that ##f## is ##R-##linear without using this property of the ##f_\alpha\,?## It is part of the set up, as we consider ##R-##modules, and thus the morphisms in this category have to be ##R-##linear. The linearity of the ##f_\alpha## extend to the linearity of ##f##, a property (of ##f\,##) which isn't used, but has to be shown (in order to stay in the category). But you will need the linearity of the ##f_\alpha## but these are given.

 

[Math Amateur]

Thanks fresh_42

Appreciate your help ...

Peter

 

[Math Amateur]

Hi fresh_42 ...

... have tried the following ..

We need to show ##f## is an ##R##-linear map (homomorphism) ...

We are given that the ##f_\alpha## are ##R##-linear maps, and we know that the projections ##\pi_\alpha## are ##R##-linear maps ...

We also know that ##\pi_\alpha f = f_\alpha## for each ##\alpha \in \Delta## ... ... ... ... ... (1)

Now ... we know that if ##f## is an ##R##-linear mapping then (1) holds true but ...

... how do you prove that f must necessarily be an ##R##-linear map ... ...... can you help ... ... ?

Peter

 

[fresh_42]

Just write down the definition: ##f(r\cdot x + s \cdot y) = \ldots ## with ##x=(x_\alpha)\; , \;y=(y_\alpha)## and ##f = \Pi_\alpha f_\alpha## and use the ##R-##linearity of all ##f_\alpha##.