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March 22, 2019, at 10:57 AM EST  Radical Page / RadicalWiki / Nearrings 
NearringsProblems proposed by S. Veldsman(1) A condition on a class of algebras that plays a useful role in general radical theory is the condition $(F).$ In the variety of associative rings, it is wellknown that a class of rings $\mathcal{M}$ which satisfies condition $(F)$ must consist of quasisemiprime rings (A.D. Sands, On ideals in overrings. Publ. Math. (Debrecen) 35 (1988), 274  279). Recall, condition $(F)$ imposed on the class $\mathcal{M}$ means: Whenever $J\lhd I\lhd A$ and $I/J\in \mathcal{M},$ then $J\lhd A.$ A quasisemiprime ring $A$ is a ring that satisfies $AxA=0$ $(x\in A)$ implies $x=0$ (which is equivalent to: $xA=0$ or $Ax=0$ $(x\in A)$ implies $x=0).$ A satisfactory internal characterization of such classes for nearrings is still abegging. In the variety of zerosymmetric nearrings, a partial result is known: A class $\mathcal{M}$ of nearrings in the variety of all zerosymmetric nearrings satisfies condition $(F)$ if each nearring $N$ in $\mathcal{M}$ satisfies: $xN=0$ $(x\in N)$ implies $x=0$ and whenever $\theta :I\rightarrow N$ is a surjective homomorphism with $I\lhd A,$ then $xy\in \ker \theta $ $(x,y\in I)$ implies $axay\in \ker \theta $ for all $a\in A$ (S. Veldsman, An overnilpotent radical theory for nearrings. J. Algebra 144 (1991), 248  265). One would like to replace the second requirement with some internal condition on the nearring $N$ only. What the situation is in the variety of all nearrings is still open. For some information, see S. Veldsman, On ideals and extensions of nearrings. Publ. Math. (Debrecen) 41 (1992), 13  22. (2) Filial rings have been studied extensively and their structure is wellknown (see the papers of M. Filipowicz and E.R. Puczy\l owski). A ring $A$ is a \textit{filial ring} if it satisfies: whenever $J\lhd I\lhd A,$ then $J\lhd A.$ It is known that $A$ is a filial ring if and only if for all $a\in A,$ $\left\langle a\right\rangle =\left\langle a\right\rangle ^{2}+\mathbb{Z}a$ (see A.D. Sands, On ideals in overrings. Publ. Math. (Debrecen) 35 (1988), 274  279). Characterize the filial nearrings. A partial result can be found in S. Veldsman, On ideals and extensions of nearrings. Publ. Math. (Debrecen) 41 (1992), 13  22. (3) Is a simple zerosymmetric nearring with identity equiprime? A nearring $N$ is equiprime if $anx=any$ for all $n\in N$ $(a\neq 0,a,x,y\in N)$ implies $x=y.$ (4) Is Sands' Theorem for the characterization of semisimple classes of associative rings valid in the variety of zerosymmetric nearrings? Recall, a class $\mathcal{S}$ of rings (or omegagroups) is called a semisimple class if it satisfies: $A\in \mathcal{S}\Leftrightarrow $ every nonzero ideal of $A$ has a nonzero homomorphic image in $\mathcal{S}.$ In any universal class of omegagroups, it can be shown that a class $\mathcal{S}$ is a semisimple class if and only if it is regular, coinductive, closed under extensions and satisfies the condition $(P):$ If $J\lhd I\lhd A$ and $J$ and $I$ are minimal with respect to $I/J\in \mathcal{S}$ and $A/I\in \mathcal{S},$ then $J\lhd A$ (see, for example L.C.A. van Leeuwen and R. Wiegandt, Radicals, semisimple classes and torsion theories, Acta Math. Acad. Sci. Hungar. 36 (1980), 37  47 and L.C.A. van Leeuwen and R. Wiegandt, Semisimple and torsionfree classes, Acta Math. Acad. Sci. Hungar. 38 (1981), 73  81). Sands' Theorem (see A.D. Sands, A characterization of semisimple classes. Proc. Edinburgh Math. Soc. 24 (1981), 5  7) is that in the universal class of associative rings, the equivalence holds without requiring condition $(P).$ 
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