What was the motivation to use the word "kernel" in algebra to denote the set of all arguments which are mapped to the idendity element (by a homomorphism)? Who introduced it?
KERNEL ... The use of kernel in algebra appears to be unrelated to its use in integral equations and Fourier analysis. The OED gives the following quotation from Pontrjagin’s Topological Groups i. 11 (translated by E. Lehmer 1946) "The set of all the elements of the group G which go into the identity of the group G* under the homomorphism g is called the kernel of this homomorphism."
Kern used this way (the set of elements of a group mapped to the identity by a homomorphism) is in my German-language copy of van der Waerden Algebra.
Ziegler points out some earlier uses HERE.