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<br />

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<br />

for some x_\alpha, a_\alpha<br />

... ... 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.