TITLE: What is the kernel of a homomorphism?
QUESTION [11 upvotes]: I've done a lot of problems before but I am trying to get a really basic   definition of kernel so that I may apply to any possible given question that I may be presented with.  
Would  I be correct in saying that the kernel (of a  homomorphism) is basically what I can multiply any given function by to get the identity?

REPLY [6 votes]: To the OP: the other answers have given you the definition of the kernel of a group homomorphism only, probably because you tagged this question with group theory.
If you are familiar with ring theory (if not, you will be soon), we can have a homomorphism from a ring $R$ to a ring $S$.  But a ring has both an additive identity, usually denoted $0$, and a multiplicative identity, usually denoted $1$.  In this special case, the kernel of the homomorphism is defined as the stuff in $R$ that is mapped to $0$.
Also, if you have ever taken a linear algebra course, and know about linear transformations between vector spaces, the kernel of a linear transformation is the stuff in the domain that is mapped to the $0$ vector in the co-domain.