A ring R is called symmetric, if abc = 0 implies acb = 0 for a, b, c

R. An ideal I of a ring R is called symmetric (resp. radically-symmetric) if R=I (resp. R/

) is a symmetric ring. We first show that symmetric ideals and ideals which have the insertion of factors property are radically-symmetric. We next show that if R is a semicommutative ring, then

(R) and R[x]=(

) are radically-symmetric, where (

) is the ideal of R[x] generated by

. Also we give some examples of radically-symmetric ideals which are not symmetric. Connections between symmetric ideals of R and related ideals of some ring extensions are also shown. In particular we show that if R is a symmetric (or semicommutative) (

,

)-compatible ring, then R[x;

,

] is a radically-symmetric ring. As a corollary we obtain a generalization of [13].