  For any factorization domain $\cal A$ and an algebra endomorphism $\sigma$ of
$\cal A$, there exists a non-associative algebra $({\cal
A},\sigma,[\cdot,\cdot])$ with multiplication satisfying skew-symmetry and
generalized (twisted) Jacobi identities, called a $\sigma$-deformed Witt
algebra. In this paper, we obtain the necessary and sufficient conditions for
the algebra $({\cal A},\sigma,[\cdot,\cdot])$ to be simple.
