Definition
For a morphism of rings , we define the kernel of ,
This is the additive group morphism kernel (it only cares about the additive identity of ). We also have that the kernel is a subring .
For a morphism of rings f:(R,+,⋅)→(S,+,⋅), we define the kernel of f,
kerf=f−1(0S)={r∈R∣f(r)=0S}.This is the additive group morphism kernel (it only cares about the additive identity of S). We also have that the kernel is a subring kerf≤R.