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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 \(\displaystyle R\)-module \(\displaystyle N\) and every family \(\displaystyle \{ f_\alpha \ : \ N \rightarrow M_\alpha \}_\Delta\) of \(\displaystyle R\)-linear mappings there is a unique \(\displaystyle R\)-linear mapping \(\displaystyle f \ : \ N \rightarrow \prod_\Delta M_\alpha\) ... ... "The proposition declares the family of mappings \(\displaystyle \{ f_\alpha \ : \ N \rightarrow M_\alpha \}_\Delta\) as \(\displaystyle R\)-linear mappings and also declares that \(\displaystyle f\) (see below for definition of \(\displaystyle f\)!) is an \(\displaystyle R\)-linear mapping ...

... BUT ...

I cannot see where in the proof the fact that they are \(\displaystyle R\)-linear mappings is used ...

Can someone please explain where in the proof the fact that the family of mappings \(\displaystyle \{ f_\alpha \ : \ N \rightarrow M_\alpha \}_\Delta\) and \(\displaystyle f\) are \(\displaystyle R\)-linear mappings is used ... basically ... why do these mappings have to be \(\displaystyle R\)-linear ... ?

Help will be much appreciated ...

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

View attachment 8031Hope that helps ...

Peter***EDIT***

In respect of \(\displaystyle f\) it seems we have to prove \(\displaystyle f\) is an \(\displaystyle R\)-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 \(\displaystyle R\)-module \(\displaystyle N\) and every family \(\displaystyle \{ f_\alpha \ : \ N \rightarrow M_\alpha \}_\Delta\) of \(\displaystyle R\)-linear mappings there is a unique \(\displaystyle R\)-linear mapping \(\displaystyle f \ : \ N \rightarrow \prod_\Delta M_\alpha\) ... ... "The proposition declares the family of mappings \(\displaystyle \{ f_\alpha \ : \ N \rightarrow M_\alpha \}_\Delta\) as \(\displaystyle R\)-linear mappings and also declares that \(\displaystyle f\) (see below for definition of \(\displaystyle f\)!) is an \(\displaystyle R\)-linear mapping ...

... BUT ...

I cannot see where in the proof the fact that they are \(\displaystyle R\)-linear mappings is used ...

Can someone please explain where in the proof the fact that the family of mappings \(\displaystyle \{ f_\alpha \ : \ N \rightarrow M_\alpha \}_\Delta\) and \(\displaystyle f\) are \(\displaystyle R\)-linear mappings is used ... basically ... why do these mappings have to be \(\displaystyle R\)-linear ... ?

Help will be much appreciated ...

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

View attachment 8031Hope that helps ...

Peter***EDIT***

In respect of \(\displaystyle f\) it seems we have to prove \(\displaystyle f\) is an \(\displaystyle R\)-linear mapping ... but then ... where is this done ...

Last edited: