Simple Deformed Witt Algebras

For any factorization domain $\mathcal{A}$ and an algebra endomorphism $\sigma$ of $\mathcal{A}$, 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 $(\mathcal{A},\sigma ,[\cdot ,\cdot ])$ to be simple.