Free Modules: Bland Corollary 2.2.4 - Issue on Finite Generation

[Math Amateur]

I am reading Paul E. Bland's book, "Rings and Their Modules".

I am trying to understand Section 2.2 on free modules and need help with the proof of Corollary 2.2.4.

Corollary 2.2.4 and its proof read as follows:

In the second last paragraph of Bland's proof above we read:

" ... ... If $(a_\alpha) \in R^{ ( \Delta ) }$, then $\sum_\Delta x_\alpha a_\alpha \in F$ ... ... "My question is as follows:

How, exactly, do we know that $(a_\alpha) \in R^{ ( \Delta ) }$ implies that $\sum_\Delta x_\alpha a_\alpha \in F$ ... ... that is, is it possible that for some $(a_\alpha) \in R^{ ( \Delta ) }$ there is no element $x$ such that $x = \sum_\Delta x_\alpha a_\alpha \in F$?To make sure my question is clear ... ...

If F is a free R-module with basis $\{ x_\alpha \}_\Delta$, then every element $x \in F$ can be expressed (generated) as a sum of the form:

$x = \sum_\Delta x_\alpha a_\alpha$

... ... BUT ... ... does this mean that for any element $(a_\alpha) \in R^{ ( \Delta ) }$ there is actually an element $x \in F$ such that $x = \sum_\Delta x_\alpha a_\alpha$?

... OR ... to put it another way ... could it be that for some element $(a_\alpha) \in R^{ ( \Delta ) }$ there is actually NO element $x \in F$ such that $x = \sum_\Delta x_\alpha a_\alpha$?

Can someone please clarify this issue for me?

Peter
***NOTE***I thought I would try to clarify just exactly why I am perplexed about the nature of the generation of a module or submodule by a set $S$.

Bland defines the generation of a submodule of $N$ of an $R$-module $M$ as follows:

Now consider a submodule $L$ of $M$ such that $L \subset N$.

See Figure $1$ as follows:

Now $L$, like $N$, will (according to Bland's definition) also be generated by $S$, since every element $y \in L$ will be able to be expressed as a sum

$y = \sum_{\Delta} x_\alpha a_\alpha$

where $x_\alpha \in S$ and $a_\alpha \in R$

This is possible since every element of $N$ (and hence $L$) can be expressed this way.However ... ... if we consider $x \in N$ such that $x \notin L$ then

$x = \sum_{\Delta} x_\alpha a_\alpha$

for some $x_\alpha, a_\alpha$

... ... BUT ... ... in this case, there is no $(a_\alpha) \in R^{ ( \Delta ) }$ such that

$\sum_{\Delta} x_\alpha a_\alpha \in L$

... ... BUT ... ... this is what is assumed in Bland's proof of Corollary $2.2.4$?

Can someone please clarify this issue ...

Peter

 

[mathwonk]

"If F is a free R-module with basis {xα}Δ, then every element x∈F can be expressed (generated) as a sum of the form:

x=∑Δxαaα

... ... BUT ... ... does this mean that for any element (aα)∈R(Δ) there is actually an element x∈F such that x=∑Δxαaα?

Read more: https://www.physicsforums.com "

the answer is yes, and this is just the meaning of a module. I.e. a module is closed under (finite) linear combinations. So since the basis elements do belong to the module, F, so also does any finite linear combination.