An associative ring R is said to be filial if every ideal J of an ideal I of R is an ideal in R \((J\triangleleft I\triangleleft R\Rightarrow J\triangleleft R)\). The following conditions are equivalent: (i) R is filial; (ii) For every element \(a\in R\), \((a)_ R=(a)^ 2_ R+Za\); (iii) If \(I\triangleleft R\) and S is a subring of I then \(I^ 2+S\triangleleft R\); (iv) If \(I\triangleleft R\) and S is a subring of I then for every integer \(n\geq 2\), \(I^ n+S\triangleleft R\); (v) There exists an integer \(n\geq 2\) such that for every \(I\triangleleft R\) and each subring S of I, \(I^ n+S\triangleleft R\); (vi) If \(I\triangleleft R\) and S is a subring of I then for some integer \(n\geq 2\), \(I^ n+S\triangleleft R\) (Th. 1). The last part of the paper contains some more specific results. Among others it is shown, that \(R\oplus R\) is filial iff \([a]_ R=[a]^ 2_ R\) for every \(a\in R\) (here \([a]_ R=aR+Ra+RaR)\).