  We introduce hom-associative Ore extensions as non-unital, non-associative
Ore extensions with a hom-associative multiplication, and give some necessary
and sufficient conditions when such exist. Within this framework, we construct
families of hom-associative quantum planes, universal enveloping algebras of a
Lie algebra, and Weyl algebras, all being hom-associative generalizations of
their classical counterparts, as well as prove that the latter are simple. We
also provide a way of embedding any multiplicative hom-associative algebra into
a multiplicative, weakly unital hom-associative algebra, which we call a weak
unitalization.
