Help with a proof of Sharp's Steps in commutative algebra

Asked Jul 06 '13 at 18:46

Active Jul 06 '13 at 18:46

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I'm trying to understand the following part of this book

I couldn't prove that $aR=\Pi_{i=1}^sRp^{t_i}_i$.

$\subset$ part:

$x\in aR\implies x=ar_1\implies x=up_1^{t_1}...p_s^{t_s}r_1\implies x=(ur_1)p_1^{t_1}...p_s^{t_s}, r_1\in R$, since $R$ is commutative, then $x=(ur_1p_1^{t_1})(1p_2^{t_2})...(1p_s^{t_s}x) \in \Pi_{i=1}^sRp^{t_i}_i$

I need help with the converse.

Thanks a lot

asked Jul 06 '13 at 18:46

user75086

@YACP what is RHS? – user75086 Jul 06 '13 at 19:12
There is a problem, how can I find the unit $u$ to get an element of $aR$? – user75086 Jul 06 '13 at 19:15
2

$x\in\Pi_{i=1}^sRp^{t_i}i$ then $x=r\Pi{i=1}^sp^{t_i}i$ so $x=(u^{-1}r)u\Pi{i=1}^sp^{t_i}_i=(u^{-1}r)a\in Ra$. – Jul 06 '13 at 20:09
@YACP of course, so stupid question, thank you very much, I'm very grateful for your help. – user75086 Jul 06 '13 at 20:24

Help with a proof of Sharp's Steps in commutative algebra

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