groups. It is the preimage of the zero ideal , that is, the subset of consisting of all those elements of that are mapped by to the element .The kernel is usually denoted (or a variation).In symbols:
:

Since a ring homomorphism preserves zero elements, the zero element of must belong to the kernel.The homomorphism is injective if and only if its kernel is only the singleton set .This is always the case if is a field, and is not the zero ring.