Question about kernel and homomorphism I was wondering is there any reason we take the identity e` for the kernel for ring homomorphism to be the additive identity instead of the multiplicative one?

Here's a line of reasoning that leads to this definition of kernel: among groups, one has the isomorphism $G/\text{ker}\varphi\cong \text{im}\varphi$, where $\varphi$ is any homomorphism out of $G$. This is a constantly useful theorem, so we'd like something equivalent for rings. And it's the usual definition of kernel as the inverse image of zero that gets us this theorem. As commenters have mentioned, this rests on the fact that the kernel is an ideal, so that $R/\text{ker}\varphi$ is at least a ring; in contrast the inverse image of the multiplicative identity is just some multiplicative submonoid of $R$, so that the best kind of isomorphism theorem you could get out such a "kernel" would be an isomorphism of monoids. And that's not very useful.
