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- Jun 22, 2012

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in the book, Sharp: Steps in Commutative Algebra, in Chapter 2 on ideals on page 29 we find Exercise 2.29 which reads as follows: (see attachment)

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Let \(\displaystyle R\) be a commutative ring and let \(\displaystyle m \in \mathbb{N}\).

Describe the ideal \(\displaystyle (x_1, x_2, ... \ ... ,x_n)^m \) of the ring \(\displaystyle R[x_1, x_2, ... \ ... ,x_n] \) of polynomials over R in indeterminates \(\displaystyle x_1, x_2, ... \ ... ,x_n \).

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Peter

Note: On pages 28 and 29 of Sharp we have the following relevant information: (see attachment)

"we can unambiguously define the product \(\displaystyle {\prod}_{i=1}^{n} I_i \) of ideals \(\displaystyle I_1, I_2, ... \ ... ,I_n \) of \(\displaystyle R\): we have

\(\displaystyle {\prod}_{i=1}^{n} I_i = I_1I_2 ... \ ... I_n = RL \) ... ... (1)

where

\(\displaystyle L = \{a_1, a_2, ... \ ... , a_n \ | \ a_1 \in I_1, a_2 \in I_2, ... \ ... a_n \in I_n \} \)

We therefore see that a typical element of \(\displaystyle I_1I_2 ... \ ... I_n \) is a sum of finitely many elements of L."

Note however that I have some trouble with reconciling the last statement: "a typical element of \(\displaystyle I_1I_2 ... \ ... I_n \) is a sum of

BUT ... from one it appears to me that an element of RL would be of the form \(\displaystyle r (a_1, a_2, ... \ ... , a_n) \) where \(\displaystyle r \in R\) - however this is not a finite sum ...

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Let \(\displaystyle R\) be a commutative ring and let \(\displaystyle m \in \mathbb{N}\).

Describe the ideal \(\displaystyle (x_1, x_2, ... \ ... ,x_n)^m \) of the ring \(\displaystyle R[x_1, x_2, ... \ ... ,x_n] \) of polynomials over R in indeterminates \(\displaystyle x_1, x_2, ... \ ... ,x_n \).

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**Can someone please help me get started on this problem?**Peter

Note: On pages 28 and 29 of Sharp we have the following relevant information: (see attachment)

"we can unambiguously define the product \(\displaystyle {\prod}_{i=1}^{n} I_i \) of ideals \(\displaystyle I_1, I_2, ... \ ... ,I_n \) of \(\displaystyle R\): we have

\(\displaystyle {\prod}_{i=1}^{n} I_i = I_1I_2 ... \ ... I_n = RL \) ... ... (1)

where

\(\displaystyle L = \{a_1, a_2, ... \ ... , a_n \ | \ a_1 \in I_1, a_2 \in I_2, ... \ ... a_n \in I_n \} \)

We therefore see that a typical element of \(\displaystyle I_1I_2 ... \ ... I_n \) is a sum of finitely many elements of L."

Note however that I have some trouble with reconciling the last statement: "a typical element of \(\displaystyle I_1I_2 ... \ ... I_n \) is a sum of

__of L." with the equation (1) above.__**finitely many elements**BUT ... from one it appears to me that an element of RL would be of the form \(\displaystyle r (a_1, a_2, ... \ ... , a_n) \) where \(\displaystyle r \in R\) - however this is not a finite sum ...

**Can someone please clarify this issue for me ... as well as help get a start on the problem ...**
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