Let R be a prime ring with extended centroid F and δ an F-algebraic derivation of R with the associated inner derivation ad(b). Set , the subring of constants of δ in R. We characterize the primeness and semiprimeness of R (δ) by the minimal polynomial μ(λ) of b over F. We also show that, if R (δ) is a prime PI-ring, then PI-deg(R) = PI-deg(R (δ)) × deg F b, where PI-deg(R) denotes the PI-degree of R. This is also extended to the case when R (δ) is a semiprime ring.