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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:
View attachment 6465
View attachment 6466In 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 ...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:
https://www.physicsforums.com/attachments/6467
The First Isomorphism Theorem reads as follows:
https://www.physicsforums.com/attachments/6468
I need some help with the proof of Proposition 6.2.6 ... ... ...
Proposition 6.2.6 and its proof read as follows:
View attachment 6465
View attachment 6466In 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 ...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:
https://www.physicsforums.com/attachments/6467
The First Isomorphism Theorem reads as follows:
https://www.physicsforums.com/attachments/6468