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- Jun 22, 2012

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I am reading R.Y. Sharp's book "Steps in Commutative Algebra"

At the moment I am trying to achieve a full understanding of the mechanics and nature of LEMMA 1.11 and am reflecting on Exercise 1.12 which follows it.

LEMMA 1.11 reads as follows: (see attachment)

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Let \(\displaystyle S \) be a subring of the ring \(\displaystyle R \), and let \(\displaystyle \Gamma \) be a subset of R. Then \(\displaystyle S[ \Gamma ] \) is defined to be the intersection of all subrings of R which contain both \(\displaystyle S \) and \(\displaystyle \Gamma \). (There certainly is one such subring, namely \(\displaystyle R \) itself).

Thus \(\displaystyle S[ \Gamma ] \) is a subring of R which contains both \(\displaystyle S \) and \(\displaystyle \Gamma \), and it is the smallest such subring of R in the sense that it is contained in every other subring of R that contains both \(\displaystyle S \) and \(\displaystyle \Gamma \).

In the special case in which \(\displaystyle \Gamma \) is a finite set \(\displaystyle \{ \alpha_1, \alpha_2, ... \ ... , \alpha_n \} \), we write \(\displaystyle S[ \Gamma ] \) as \(\displaystyle S[\alpha_1, \alpha_2, ... \ ... , \alpha_n] [ \).

In the special case in which \(\displaystyle S \) is commutative, and \(\displaystyle \alpha \in R \) is such that \(\displaystyle \alpha s = s \alpha \) for all \(\displaystyle s \in S \) we have

\(\displaystyle S[ \alpha ] = \{ {\sum}_{i=0}^{t} s_i {\alpha}^i \ : \ t \in \mathbb{N}_0, s_1, s_2, ... \ ... , s_t \in S \} \)

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Now a couple of issues/problems ...

Issue/Problem 1

Given that \(\displaystyle S[ \Gamma ] \) is the intersection of all subrings of \(\displaystyle R \) which contain both \(\displaystyle S \) and \(\displaystyle \Gamma \), it should be equal to the subring generated by the union of \(\displaystyle S \) and \(\displaystyle \Gamma \) [Dummit and Foote establish this equivalence for ideals in Section 7.4 page 251 - so it should work for subrings]. Similarly, \(\displaystyle S[ \alpha ] \) (the same situation restricted to one variable) should be equal to the subring generated by \(\displaystyle S\) and \(\displaystyle \alpha \).

So the subring \(\displaystyle S[ \alpha ] \) should contain all finite sums of terms of the form \(\displaystyle s_i \alpha^i , i = 1, 2, ... \). But we can write \(\displaystyle s^i = s_i \) for some element \(\displaystyle s_i \in S \) since \(\displaystyle S\) is a subring. Therefore the terms in \(\displaystyle S[ \alpha ] \) can be expressed \(\displaystyle \sum s_i \alpha^i \)

Problem/Issue 2

I am trying to make a start on Exercise 1.12 which follows and is related to LEMMA 1.11, but not making any significant headway ...

Exercise 1.12 reads as follows: (see attachment)

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Let S be a subring of the commutative ring R, and let \(\displaystyle \Gamma, \Delta \) be subsets of R

Show that \(\displaystyle S[\Gamma \cup \Delta] = S[\Gamma] [\Delta] \) and

\(\displaystyle S[\Gamma] = \underset{\Omega \subseteq \Gamma , \ | \Omega | \lt \infty}{\bigcup} S[ \Omega] \)

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Can someone help me to make a significant start on this exercise?

Would appreciate some help

Peter

At the moment I am trying to achieve a full understanding of the mechanics and nature of LEMMA 1.11 and am reflecting on Exercise 1.12 which follows it.

LEMMA 1.11 reads as follows: (see attachment)

---------------------------------------------------------------------------------------

Let \(\displaystyle S \) be a subring of the ring \(\displaystyle R \), and let \(\displaystyle \Gamma \) be a subset of R. Then \(\displaystyle S[ \Gamma ] \) is defined to be the intersection of all subrings of R which contain both \(\displaystyle S \) and \(\displaystyle \Gamma \). (There certainly is one such subring, namely \(\displaystyle R \) itself).

Thus \(\displaystyle S[ \Gamma ] \) is a subring of R which contains both \(\displaystyle S \) and \(\displaystyle \Gamma \), and it is the smallest such subring of R in the sense that it is contained in every other subring of R that contains both \(\displaystyle S \) and \(\displaystyle \Gamma \).

In the special case in which \(\displaystyle \Gamma \) is a finite set \(\displaystyle \{ \alpha_1, \alpha_2, ... \ ... , \alpha_n \} \), we write \(\displaystyle S[ \Gamma ] \) as \(\displaystyle S[\alpha_1, \alpha_2, ... \ ... , \alpha_n] [ \).

In the special case in which \(\displaystyle S \) is commutative, and \(\displaystyle \alpha \in R \) is such that \(\displaystyle \alpha s = s \alpha \) for all \(\displaystyle s \in S \) we have

\(\displaystyle S[ \alpha ] = \{ {\sum}_{i=0}^{t} s_i {\alpha}^i \ : \ t \in \mathbb{N}_0, s_1, s_2, ... \ ... , s_t \in S \} \)

------------------------------------------------------------------------------------

Now a couple of issues/problems ...

Issue/Problem 1

Given that \(\displaystyle S[ \Gamma ] \) is the intersection of all subrings of \(\displaystyle R \) which contain both \(\displaystyle S \) and \(\displaystyle \Gamma \), it should be equal to the subring generated by the union of \(\displaystyle S \) and \(\displaystyle \Gamma \) [Dummit and Foote establish this equivalence for ideals in Section 7.4 page 251 - so it should work for subrings]. Similarly, \(\displaystyle S[ \alpha ] \) (the same situation restricted to one variable) should be equal to the subring generated by \(\displaystyle S\) and \(\displaystyle \alpha \).

So the subring \(\displaystyle S[ \alpha ] \) should contain all finite sums of terms of the form \(\displaystyle s_i \alpha^i , i = 1, 2, ... \). But we can write \(\displaystyle s^i = s_i \) for some element \(\displaystyle s_i \in S \) since \(\displaystyle S\) is a subring. Therefore the terms in \(\displaystyle S[ \alpha ] \) can be expressed \(\displaystyle \sum s_i \alpha^i \)

**Can someone please confirm that the above reasoning is sound ... or not ...**Problem/Issue 2

I am trying to make a start on Exercise 1.12 which follows and is related to LEMMA 1.11, but not making any significant headway ...

Exercise 1.12 reads as follows: (see attachment)

------------------------------------------------------------------------------------

Let S be a subring of the commutative ring R, and let \(\displaystyle \Gamma, \Delta \) be subsets of R

Show that \(\displaystyle S[\Gamma \cup \Delta] = S[\Gamma] [\Delta] \) and

\(\displaystyle S[\Gamma] = \underset{\Omega \subseteq \Gamma , \ | \Omega | \lt \infty}{\bigcup} S[ \Omega] \)

-------------------------------------------------------------------------------------

Can someone help me to make a significant start on this exercise?

Would appreciate some help

Peter

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