The purpose of this paper is to prove the following results; (1) Let R be a prime ring of char $(R) eq 2$ and I a nonzero left ideal of R. The existence of a nonzero symmetric bi-derivation D : $R imesR;longrightarrow;$ such that d is sew-commuting on I where d is the trace of D forces R to be commutative (2) Let m and n be integers with $m; eq;0.;or;n eq;0$. Let R be a noncommutative prime ring of char$ (R)) eq ; 2-1; p_1 ;n_1$ where p is a prime number which is a divisor of m, and I a nonzero two-sided ideal of R. Let $D_1$ ; $R; imes;R;longrightarrow;and;$ $D_2;:;R; imes;R;longrightarrow;R$ be symmetric bi-derivations. Suppose further that there exists a symmetric bi-additive mapping B ; $R; imes;R;longrightarrow;and;$ such that $md_1(chi)chi ＋ nchi d_2(chi)=f(chi$) holds for all $chi$$in$I, where $d_1 ;and; d_2$ are the traces of $D_1 ;and; D_2$ respectively and f is the trace of B. Then we have $D_1=0 ;and; D_2=0$.