I am generally unsure of what a kernel is in abstract algebra. From my background in linear algebra, I understand that the kernel is the same as the null space (I learned it as the null space).
For example, if I have some homomorphism $f: (\mathbb{R},+) \longrightarrow (\mathbb{C}^*, \cdot)$, how would I go about describing the kernel?
Ther kernel of a (group) homomorphism is the set of all elements that get mapped onto the neutral element.
So $\ker(f)=\{x\in\mathbb{R}: f(x)=1\}$, as $1$ is the neutral element of $\mathbb{C}^\ast$ with regards to multiplication.
How to describe the kernel then depends clearly on the specific homomorphism.