I don't understand this page, https://www.proofwiki.org/wiki/Kernel_of_Ring_Homomorphism_is_Subring, but how can this be a true statement? Wouldn't a ring morphism map the multiplicitive identity to itself? So it wouldn't be in the kernel, so how could the kernel be a subring? I happened upon this whilst trying to figure out why the kernel of a morphism is an ideal in the pre-image ring or whatever. Anyone enlighten me and alleviate my confusion?