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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 - 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
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