I am trying to represent myself quotient groups and I'm having trouble seeing what the kernel of a homomorphism : $\Phi: G \rightarrow G/H$ is (be it a ring homomorphism or a group homomorphism).
I understand that $\mathbb{Z}/n\mathbb{Z}$ is a quotient group that is cyclic and that the kernel of the application: $\Pi : \mathbb{Z} \rightarrow \mathbb{Z}/n\mathbb{Z}$ are all the elements of $n\mathbb{Z}$.
But in the more general case I have trouble truly visualizing the kernel of a quotient group:
Let us take the 2 following examples:
In the case of a group homomorphism, Let G be a group and H be a subgroup of G: $\Xi : G \rightarrow G/H$, $\Xi(g) \mapsto gH$ I know that its kernel is H but I don't understand why.
In the case of a ring homomorphism: If $(A,+,*)$ is a ring and $I$ is a two-sided ideal of $A$. Let the ring homomorphism $\chi: A \rightarrow A/I$, $\chi(a) \mapsto a+I$ I know that its kernel is I but I don't understand why.
Thank you for any clarification