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I am reading Stephen Lovett's book, "Abstract Algebra: Structures and Applications" and am currently focused on Section 6.2: Rings of Fractions ...
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:
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: