  A U(1) BF-Yang-Mills theory on noncommutative ${\mathbb{R}}^4$ is presented
and in this formulation the U(1) Yang-Mills theory on noncommutative
${\mathbb{R}}^4$ is seen as a deformation of the pure BF theory. Quantization
using BRST symmetry formalism is discussed and Feynman rules are given.
Computations at one-loop order have been performed and their renormalization
studied. It is shown that the U(1) BFYM on noncommutative ${\mathbb{R}}^4$ is
asymptotically free and its UV-behaviour in the computation of the
$\beta$-function is like the usual SU(N) commutative BFYM and Yang Mills
theories.
