Consider the homomorphism between a field $K$ and $A$ which is finitely generated $K$-algebra (that is it can be written $K[x_1,...,x_n]$). If $\phi: A \to K$ is a homomorphism between $K$-algebras does it always hold that $A=K+\operatorname{Ker}(\phi)$?
I have never encountered this rule so what are sufficient conditions for this to hold?
For all $a\in A$ we can write $a=\underbrace{\phi(a)}_{\in\,K}+\underbrace{a-\phi(a)}_{\in\,\ker\phi}$.
More generally if $\phi$ is a surjective map out of $R$, and its restriction to the subset $S\subseteq R$ is also onto, then we can write $R=S+\ker\phi$ by the same reasoning. A similar proposition holds for groups, where $\phi$ is surjective out of $G$ and also onto restricted to a subset $X\subseteq G$, then $G=X\ker\phi$.