Abstract : A ring $R$ is called symmetric, if $abc=0$ implies $acb=0$ for $a,b,c\in R$. An ideal $I$ of a ring $R$ is called symmetric (resp. radically-symmetric) if $R/I$ (resp. $ R/\sqrt{I}$) 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 $T_n(R)$ and $ R[x]/(x^n)$ are radically-symmetric, where $(x^n)$ is the ideal of $R[x]$ generated by $x^n$. 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) $(\alpha,\delta)$-compatible ring, then $R[x;\alpha,\delta]$ is a radically-symmetric ring. As a corollary we obtain a generalization of [13].