I am reading Stephen Lovett's book, "Abstract Algebra: Structures and Applications" and am currently focused on Chapter 5 ...(adsbygoogle = window.adsbygoogle || []).push({});

I need some help with Example 5.2.1 in Section 5.2: Rings Generated by Elements ...

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(precisely in Lovett's words) the ring ##\mathbb{Z} [ \frac{1}{2} ]## ... ...actually

... ... according to his introduction which I quoted Lovett says that the ring ##\mathbb{Z} [ \frac{1}{2} ]## is thesubring of ##\mathbb{Q}## that contains ##\mathbb{Z}## and ##\frac{1}{2}## ... ...smallest (by inclusion)

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

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