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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:View attachment 3533
View attachment 3534In 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
***EDIT***
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:View attachment 3535Now consider a submodule $$L$$ of $$M$$ such that $$L \subset N$$.
See Figure $$1$$ as follows:https://www.physicsforums.com/attachments/3536Now $$L$$, 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
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:View attachment 3533
View attachment 3534In 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
***EDIT***
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:View attachment 3535Now consider a submodule $$L$$ of $$M$$ such that $$L \subset N$$.
See Figure $$1$$ as follows:https://www.physicsforums.com/attachments/3536Now $$L$$, 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
Last edited: