I am reading Stephen Lovett's book, "Abstract Algebra: Structures and Applications" and am currently focused on Section 6.2: Rings of Fractions ...(adsbygoogle = window.adsbygoogle || []).push({});

I need some help with the proof of Proposition 6.2.6 ... ... ...

Proposition 6.2.6 and its proof read as follows:

In the above proof by Lovett we read the following:

" ... ... By Lemma 6.2.5, the function ##\phi## is injective, so by the First Isomorphism Theorem, ##R## is isomorphic to ##\text{Im } \phi##. ... ... "

*** NOTE *** The function ##\phi## is defined in Lemma 6.2.5 which I have provided below ... ..

My questions are as follows:

Question 1

I am unsure of exactly how the First Isomorphism Theorem establishes that ##R## is isomorphic to ##\text{Im } \phi##.

Can someone please show me, rigorously and formally, how the First Isomorphism Theorem applies in this case ...

Question 2

I am puzzled as to why the First Isomorphism Theorem is needed in the first place as ##\phi## is an injection by Lemma 6.2.5 ... and further ... obviously the map of ##R## to ##\text{Im } \phi## is onto, that is a surjection ... so ##R## is isomorphic to ##\text{Im } \phi## ... BUT ... why is Lovett referring to the First Isomorphism Theorem ... I must be missing something ... hope someone can clarify this issue ...

Hoe that someone can help ... ...

Peter

===================================================

In the above, Lovett refers to Lemma 6.2.5 and the First Isomorphism Theorem ... so I am providing copies of both ...

Lemma 6.2.5 reads as follows:

The First Isomorphism Theorem reads as follows:

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# I Rings of Fractions ... Lovett, Section 6.2, Proposition 6.2

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