Strongly filial rings

  • D. I. C. Mendes
Original Paper


A subclass of the class of filial rings is studied. A ring R is defined to be strongly filial if, for every element a of R, \(\left( a\right) _{R}=\left( a^{2}\right) _{R}+\mathbb {Z}a\), where \(\left( a\right) _{R}\) and \(\left( a^{2}\right) _{R}\) denote the principal ideal of R generated by a and the principal ideal of R generated by \(a^{2}\), respectively, and \( \mathbb {Z}\) denotes the set of integers. It is proved that R is strongly filial if and only if every subring of R, which is both left and right accessible in R, is an ideal of R. Several other characterisations and properties of these rings are presented, motivated by known analogous results for filial and left filial rings.


Accessible subrings Ideals Strongly f-regular ring  Semiprime ring Prime radical 

Mathematics Subject Classification

16D25 16D80 



This research was supported by FEDER and Portuguese funds through the Centre for Mathematics (University of Beira Interior) and the Portuguese Foundation for Science and Technology (FCT— Fundação para a Ciência e a Tecnologia), Project PEst-OE/MAT/UI0212/2013.


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Copyright information

© The Managing Editors 2015

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceUniversity of Beira InteriorCovilhãPortugal

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