Rings Generated by Elements - Lovett, Example 5.2.1 .... ....

  • #1
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I am reading Stephen Lovett's book, "Abstract Algebra: Structures and Applications" and am currently focused on Chapter 5 ...

I need some help with Example 5.2.1 in Section 5.2: Rings Generated by Elements ...
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In the Introduction to Section 5.2.1 (see text above) Lovett writes:

" ... ... R[S] denotes the smallest (by inclusion) subring of ##A## that contains both ##R## and ##S## ... ... "Then, a bit later, in Example 5.2.1 concerning the ring ##\mathbb{Z} [ \frac{1}{2} ]## Lovett writes:" ... ... It is not hard to show that the set##\{ \frac{k}{ 2^n} \ | \ k.n \in \mathbb{Z} \}##is a subring of ##\mathbb{Q}##. Hence, this set is precisely the ring ##\mathbb{Z} [ \frac{1}{2} ]## ... ... ... "BUT ...

How has Lovett actually shown that the set ##\{ \frac{k}{ 2^n} \ | \ k.n \in \mathbb{Z} \}## as a subring of \mathbb{Q} is actually (precisely in Lovett's words) the ring ##\mathbb{Z} [ \frac{1}{2} ]## ... ...

... ... according to his introduction which I quoted Lovett says that the ring ##\mathbb{Z} [ \frac{1}{2} ]## is the smallest (by inclusion) subring of ##\mathbb{Q}## that contains ##\mathbb{Z}## and ##\frac{1}{2}## ... ...Can someone please explain to me exactly how Lovett has demonstrated this ... ...

... and ... if Lovett has not clearly proved this can someone please demonstrate a proof ...Just one further clarification ... is Lovett here dealing with ring extensions ... ... ?

Hope someone can help ...

Peter
 

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  • #2
1) {k/2n | k,n∈ℤ} is a subring that contains ℤ and 1/2. Therefore, ℤ[1/2] ⊆ {k/2n | k,n∈ℤ}.
2) Pick any element k/2n, k,n∈ℤ. For ℤ[1/2] to be a closed ring under an obvious series of operations, it must contain that element. Therefore {k/2n | k,n∈ℤ} ⊆ ℤ[1/2].
 
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Thanks FactChecker ... I think I now follow ...

Appreciate your help ...

Peter
 
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