When i show that any map is injective,

Question

In group theory, When i show that homomorphic mapping $\phi$ which maps group $A$ to group $B$ is injective, It is enough to show $ker\phi =${$e$} which $e$ is an identity element of $A$.

But Ring, Field or any algebraic space with at least two operations and identities (e.g 0 and 1), there is a question about identity $e$.

Is identity $e$ for first operation or for second one ?

Answer 1

For the examples you've listed, the kernel is the set of objects which are mapped to the zero element, and accordingly being injective is equivalent to having $\ker\phi=\{0\}$.