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A ring  R  is called filial (left weakly filial) if it satisfies the following condition: for all subrings  J  and  I  of  R , if  J  is an ideal of  I  and  I  is an ideal of  R , then  J  is an ideal (left ideal) of  R . If every subring of  R  is filial (left weakly filial), then  R  is called fully filial (fully left weakly filial). In this paper, some new characterizations of left weakly filial rings in terms of the prime radical are given. Left weakly filial rings that are matrix rings, polynomial rings and the direct sum of copies of a ring are also investigated. Recently, in [3], [4] and [13], classification theorems for semiprime torsion fully filial rings and for torsion-free fully filial rings were presented. We study fully left weakly filial rings and show that semiprime torsion rings and semiprime torsion-free rings are fully left weakly filial if and only if they are fully filial.

On rings related to filial and left filial rings