Adjoining an identity to a finite filial ring. (Q2512560)

The following main fact is proved. Let \(R\) be an associative filial ring, \(R_p=\{x\in R;\;p^kx=0\) for some \(k\in\mathbb N\}\). If \(|R_p |\leq p^3\) for each prime number \(p\), then \(R\) is an ideal in some filial ring \(S\) with an identity. For every prime number \(p\), there exists a commutative filial ring \(I\) of order \(p^4\) such that \(p^2I=(0)\) and \(I\) is not an ideal in any filial ring with an identity.