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I am reading Paul E. Bland's book, "Rings and Their Modules".
I am trying to understand Chapter 4, Section 4.1 on generating and cogenerating classes and need help with the proof of Proposition 4.1.1.
Proposition 4.1.1 and its proof read as follows:
View attachment 3649
View attachment 3650
I need some help with what seems a fairly intuitive step in the logic of the proof of $$(3) \Longrightarrow (4)$$ - see text above.In the proof of $$(3) \Longrightarrow (4)$$ Bland writes:
" ... ... Thus, by (3) there is a finite set $$F \subseteq \Delta$$ and an epimorphism $$ \phi \ : \ R^{ ( F ) } \longrightarrow M$$.
If we let
$$F = \{ 1,2, \ ... \ ... \ ,n \}$$
and if
$$\{ e_i \}_{i =1}^n$$
is the canonical basis for the free $$R$$-module $$R^{ ( n ) }$$, then the finite set
$$X = \{ \phi (e_i) \}_{i =1}^n$$
will generate $$M$$. ... ... "My question is the following:
Why, exactly, if $$\{ e_i \}_{i =1}^n$$ is the canonical basis for the free $$R$$-module $$R^{ ( n ) }$$, are we guaranteed that $$X$$ will generate $$M$$? [this does seem intuitive - but why EXACTLY! ]I would really appreciate some help with this issue.Peter***EDIT***
I now have a second question:
In the above text, Bland writes:
" ... ... Every R-module is the homomorphic image of a free R-module, ... ... "
So if that is true, then we have a set $$\Delta$$ and a homomorphism $$R^{ ( \Delta ) } \longrightarrow M$$ ... ...
... ... BUT ... ... Bland claims we have an epimorphism $$R^{ ( \Delta ) } \longrightarrow M$$ ...
How do we know that we not only have a homomorphism, but that we have an epimorphism?
Hep will be appreciated ...
I am trying to understand Chapter 4, Section 4.1 on generating and cogenerating classes and need help with the proof of Proposition 4.1.1.
Proposition 4.1.1 and its proof read as follows:
View attachment 3649
View attachment 3650
I need some help with what seems a fairly intuitive step in the logic of the proof of $$(3) \Longrightarrow (4)$$ - see text above.In the proof of $$(3) \Longrightarrow (4)$$ Bland writes:
" ... ... Thus, by (3) there is a finite set $$F \subseteq \Delta$$ and an epimorphism $$ \phi \ : \ R^{ ( F ) } \longrightarrow M$$.
If we let
$$F = \{ 1,2, \ ... \ ... \ ,n \}$$
and if
$$\{ e_i \}_{i =1}^n$$
is the canonical basis for the free $$R$$-module $$R^{ ( n ) }$$, then the finite set
$$X = \{ \phi (e_i) \}_{i =1}^n$$
will generate $$M$$. ... ... "My question is the following:
Why, exactly, if $$\{ e_i \}_{i =1}^n$$ is the canonical basis for the free $$R$$-module $$R^{ ( n ) }$$, are we guaranteed that $$X$$ will generate $$M$$? [this does seem intuitive - but why EXACTLY! ]I would really appreciate some help with this issue.Peter***EDIT***
I now have a second question:
In the above text, Bland writes:
" ... ... Every R-module is the homomorphic image of a free R-module, ... ... "
So if that is true, then we have a set $$\Delta$$ and a homomorphism $$R^{ ( \Delta ) } \longrightarrow M$$ ... ...
... ... BUT ... ... Bland claims we have an epimorphism $$R^{ ( \Delta ) } \longrightarrow M$$ ...
How do we know that we not only have a homomorphism, but that we have an epimorphism?
Hep will be appreciated ...
Last edited: