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I am reading Paolo Aluffi's book, Algebra: Chapter 0.
I have a question related to Aluffi's description of initial objects in the category ring ... ...
In Chapter III, Section 2. on the category ring, we read the following:View attachment 4480
View attachment 4481QUESTION 1
In the above text, we read the following:
" ... ... This ring homomorphism is unique, since it is determined by the requirement that $$\phi (1) = 1_R$$ and by the fact that $$\phi$$ preserves addition ... ... "Can someone please explain to me (precisely, rigorously and formally) why the requirement that $$\phi (1) = 1_R$$ and the fact that $$\phi$$ preserves addition imply that ring homomorphism is unique?
(Intuitively the above seems true ... but how do you show this exactly and precisely ... wonder if I am overthinking this matter ... )QUESTION 2
In the above text, we read the following:
" ... ... But $$\phi$$ is in fact a ring homomorphism, since $$\phi (1) = 1_R$$, and
$$\phi (mn) = (mn) 1_R = m(n1_R) = (m1_R) \cdot (n 1_R) = \phi (m) \cdot \phi (n)
$$
where the equality $$m(n1_R) = (m1_R) \cdot (n 1_R)$$ holds by the distributivity axiom ... ... "Can someone explain how the equality $$m(n1_R) = (m1_R) \cdot (n 1_R)$$ follows from the distributivity axiom
Help will be much appreciated ... ...
Peter
I have a question related to Aluffi's description of initial objects in the category ring ... ...
In Chapter III, Section 2. on the category ring, we read the following:View attachment 4480
View attachment 4481QUESTION 1
In the above text, we read the following:
" ... ... This ring homomorphism is unique, since it is determined by the requirement that $$\phi (1) = 1_R$$ and by the fact that $$\phi$$ preserves addition ... ... "Can someone please explain to me (precisely, rigorously and formally) why the requirement that $$\phi (1) = 1_R$$ and the fact that $$\phi$$ preserves addition imply that ring homomorphism is unique?
(Intuitively the above seems true ... but how do you show this exactly and precisely ... wonder if I am overthinking this matter ... )QUESTION 2
In the above text, we read the following:
" ... ... But $$\phi$$ is in fact a ring homomorphism, since $$\phi (1) = 1_R$$, and
$$\phi (mn) = (mn) 1_R = m(n1_R) = (m1_R) \cdot (n 1_R) = \phi (m) \cdot \phi (n)
$$
where the equality $$m(n1_R) = (m1_R) \cdot (n 1_R)$$ holds by the distributivity axiom ... ... "Can someone explain how the equality $$m(n1_R) = (m1_R) \cdot (n 1_R)$$ follows from the distributivity axiom
Help will be much appreciated ... ...
Peter
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