Kernel of a Ring Homomorphism

Let f : R → S : 𝑓 → 𝑅 𝑆 f:R\to S be a Ring Homomorphism. The kernel of f 𝑓 f , denoted ker ⁡ f kernel 𝑓 \ker f is the Two-sided Ideal ker ⁡ f ⊆ R kernel 𝑓 𝑅 \ker f\subseteq R of the subset of R 𝑅 R that lies in the Fibre over 0 : S : 0 𝑆 0:S .