I 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 $f_\alpha$.