In the proof of 3rd proposition
I can prove the intersection of all ideals is the kernel of the map, but why does it imply this proposition is true?
2026-04-15 10:33:28.1776249208
Question about some details of a proof of Chinese Remainder Theorem
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$3)$ it is a general fact and a funny excercise that if the kernel of a (ring) homomorphism is trivial (so it contains only the neutral element) then the homomorphism is injective.
How to prove it? Start by supposing there exist two elements $a,b$ in the ring such that $\phi(a)=\phi(b)$ then $\phi(a)-\phi(b)=0$. Using the fact that $\phi$ is an homomorphism can you figure out how to continue? Please note that here we use only one of the ring operation so this proof remains true when we work with groups instead of ring (the "only one" means we use the operation under which the ring is an abelian group).