MHB Free Modules - Another issue regarding Bland Proposition 2.2.4

  • Thread starter Thread starter Math Amateur
  • Start date Start date
  • Tags Tags
    Modules
Math Amateur
Gold Member
MHB
Messages
3,920
Reaction score
48
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 - that is some further help ... (for my first problem with the proof see my post "http://mathhelpboards.com/linear-abstract-algebra-14/free-modules-bland-corollary-2-2-4-issue-regarding-finite-generation-modules-13196.html" )Corollary 2.2.4 and its proof read as follows:View attachment 3538
View attachment 3539The last sentence in Bland's proof reads as follows:

" ... ... If $$ f \ : \ R^{ ( \Delta ) } \ \rightarrow \ F $$ is an isomorphism and $$f ( e_\alpha) = x_\alpha$$ then it follows that $$\{ x_\alpha \}_\Delta$$ is a basis for $$F$$. ... ... "Although the above statement seems plausible I am unable to frame a formal and rigorous proof of the statement ...

Can someone please show me exactly why $$ f \ : \ R^{ ( \Delta ) } \ \rightarrow \ F $$ is an isomorphism and $$f ( e_\alpha) = x_\alpha$$ implies that $$\{ x_\alpha \}_\Delta$$ is a basis for $$F$$?

Help will be appreciated ...

Peter
 
Physics news on Phys.org
Peter said:
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 - that is some further help ... (for my first problem with the proof see my post "http://mathhelpboards.com/linear-abstract-algebra-14/free-modules-bland-corollary-2-2-4-issue-regarding-finite-generation-modules-13196.html" )Corollary 2.2.4 and its proof read as follows:View attachment 3538
View attachment 3539The last sentence in Bland's proof reads as follows:

" ... ... If $$ f \ : \ R^{ ( \Delta ) } \ \rightarrow \ F $$ is an isomorphism and $$f ( e_\alpha) = x_\alpha$$ then it follows that $$\{ x_\alpha \}_\Delta$$ is a basis for $$F$$. ... ... "Although the above statement seems plausible I am unable to frame a formal and rigorous proof of the statement ...

Can someone please show me exactly why $$ f \ : \ R^{ ( \Delta ) } \ \rightarrow \ F $$ is an isomorphism and $$f ( e_\alpha) = x_\alpha$$ implies that $$\{ x_\alpha \}_\Delta$$ is a basis for $$F$$?

Help will be appreciated ...

Peter
First we verify that $\{x_\alpha\}_{\Delta}$ generates $F$.
Let $m\in F$. Note that $f$ is surjective.
Therefore $f(r)=m$ for some $r\in R^{\delta}$.
Now $r=\sum_{\alpha\in \Delta} r_\alpha e_\alpha$ for some $r\alpha\in R$ (where almost all of $r_\alpha$'s are $0$).
Thus, $f(r)=\sum_{\alpha\in \Delta}r_\alpha f(e_\alpha)=\sum_{\alpha \in \Delta} r_\alpha x_\alpha$.

Thus $\{x_\alpha\}_{\alpha\in \Delta}$ generates $F$.

Now we need to verify that each element of $F$ can be expressed uniquely as an $R$-linear combination of $x_\alpha$'s.

Say $\sum_{\alpha\in \Delta} r_\alpha x_\alpha=0$.
Then $\sum_{\alpha\in \Delta}r_\alpha f(e_\alpha)=0$.
Therefore $f(\sum_{\alpha\in \Delta} r_\alpha e_\alpha) =0$.
Which gives $\sum_{\alpha\in \Delta} r_\alpha e_\alpha=0$.
This leads to $r_\alpha=0$ for all $\alpha\in \Delta$.

Hence we are done.
 
Thread 'Determine whether ##125## is a unit in ##\mathbb{Z_471}##'
This is the question, I understand the concept, in ##\mathbb{Z_n}## an element is a is a unit if and only if gcd( a,n) =1. My understanding of backwards substitution, ... i have using Euclidean algorithm, ##471 = 3⋅121 + 108## ##121 = 1⋅108 + 13## ##108 =8⋅13+4## ##13=3⋅4+1## ##4=4⋅1+0## using back-substitution, ##1=13-3⋅4## ##=(121-1⋅108)-3(108-8⋅13)## ... ##= 121-(471-3⋅121)-3⋅471+9⋅121+24⋅121-24(471-3⋅121## ##=121-471+3⋅121-3⋅471+9⋅121+24⋅121-24⋅471+72⋅121##...
Back
Top