Let $R$ be a prime ring with center $Z$, $I$ a nonzero ideal of $R$, and $sigma$ a nontrivial automorphism of $R$ such that ${  (xcirc y)^{sigma} -(xcirc y)}^nin Z~~~~mbox{for all}~~x,yin I$. If either char$(R)> n$ or char $(R) = 0$, then $R$ satisfies $s_4$, the standard identity in $4$ variables.

Keywords: Prime ring, Ideal, Automorphism