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I am reading Dummit and Foote, Example 2 of Section 15.4, page 708.
Rewriting the assertions of the example as exercise style questions, EXAMPLE 2, reads as follows:
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Let R be any commutative ring with 1 and let f be any element of R. Let D be the multiplicative set $$ \{ f^n \ | \ n \ge 0\} $$ of non-negative powers of f in R.
Define $$ R_f = D^{-1}R $$.
(a) Show that $$ R_f = 0 $$ if and only if f is nilpotent.
(b) Show that if f is not nilpotent, then f becomes a unit in R
(c) Show that $$ R_f \cong R[x]/(xf -1)$$
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I can manage (a) and (b) I think - my attempts follow - but need help to get started on (c)
Attempt at (a)
$$ R_f = \{ r/d \ | \ r \in R, d \in D \} $$
where r/d = s/e if and only if x(er - ds) = 0 for some $$ x \in D $$
$$ f $$ nilpotent $$ \Longrightarrow f^m = 0 $$ for some $$ m \in Z^+ $$
$$ \Longrightarrow \ 0 \in D $$
$$ \Longrightarrow \ R_f = D^{-1}R = \{ 0 \} $$
Attempt at (b)
$$ R_f = \{ \frac{r}{f^m} \ | \ r \in R, f^m \in R_f = D^{-1}R $$
Now $$ f \in R_f \Longrightarrow \ \frac{f^1}{f^0} \in R_f \Longrightarrow \ \frac{f^1}{1} $$
But we have $$ \frac{1}{f^1} \in R_f $$
So f is a unit in $$ R_f $$ since $$ \frac{f^1}{1} \times \frac{1}{f^1} = \frac{f^1}{f^1} = 1 $$ where $$ \frac{f^1}{1} \in R_f $$ and $$ \frac{1}{f^1} \in R_f $$
Can someone please confirm that arguments (a) and (b) above are OK?
Attempt at (c)
PROBLEM ... ... I cannot get started on the third assertion above ie (c) above. Can someone please help me get started on (c) ... ...
Peter
Rewriting the assertions of the example as exercise style questions, EXAMPLE 2, reads as follows:
-------------------------------------------------------------------------------------
Let R be any commutative ring with 1 and let f be any element of R. Let D be the multiplicative set $$ \{ f^n \ | \ n \ge 0\} $$ of non-negative powers of f in R.
Define $$ R_f = D^{-1}R $$.
(a) Show that $$ R_f = 0 $$ if and only if f is nilpotent.
(b) Show that if f is not nilpotent, then f becomes a unit in R
(c) Show that $$ R_f \cong R[x]/(xf -1)$$
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I can manage (a) and (b) I think - my attempts follow - but need help to get started on (c)
Attempt at (a)
$$ R_f = \{ r/d \ | \ r \in R, d \in D \} $$
where r/d = s/e if and only if x(er - ds) = 0 for some $$ x \in D $$
$$ f $$ nilpotent $$ \Longrightarrow f^m = 0 $$ for some $$ m \in Z^+ $$
$$ \Longrightarrow \ 0 \in D $$
$$ \Longrightarrow \ R_f = D^{-1}R = \{ 0 \} $$
Attempt at (b)
$$ R_f = \{ \frac{r}{f^m} \ | \ r \in R, f^m \in R_f = D^{-1}R $$
Now $$ f \in R_f \Longrightarrow \ \frac{f^1}{f^0} \in R_f \Longrightarrow \ \frac{f^1}{1} $$
But we have $$ \frac{1}{f^1} \in R_f $$
So f is a unit in $$ R_f $$ since $$ \frac{f^1}{1} \times \frac{1}{f^1} = \frac{f^1}{f^1} = 1 $$ where $$ \frac{f^1}{1} \in R_f $$ and $$ \frac{1}{f^1} \in R_f $$
Can someone please confirm that arguments (a) and (b) above are OK?
Attempt at (c)
PROBLEM ... ... I cannot get started on the third assertion above ie (c) above. Can someone please help me get started on (c) ... ...
Peter
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