The purpose of this article is to examine zero division relations between elements and ideals of general algebras using set-theoretical tools. By splitting an algebra into subsets according to zero division relations, it is possible to establish certain zero division rules on linear combinations and thus arbitrary elements of a given algebra. We investigate in detail these subsets, properties of zero division relation, subsets of zero divisors, and annihilators in hom-algebras and in particular hom-associative algebras. For hom-associative algebras, we investigate properties of zero division relation subsets of the kernel of the twisting map, annihilators. Furthermore, we investigate kernels of right and left multiplication operators, ideals, hom-ideals, simplicity, hom-simplicity and domain properties for general hom-algebras and hom-associative algebras, and low-dimensional hom-associative algebras of nonassociative type using the kernels of twisting maps, annihilators, and their zero division relation subsets.