# Nearrings

## Problems 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 well-known that a class of rings $\mathcal{M}$ which satisfies condition $(F)$ must consist of quasi-semiprime rings (A.D. Sands, On ideals in over-rings. 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 quasi-semiprime 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 near-rings is still abegging. In the variety of zero-symmetric near-rings, a partial result is known: A class $\mathcal{M}$ of near-rings in the variety of all zero-symmetric near-rings satisfies condition $(F)$ if each near-ring $N$ 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 $x-y\in \ker \theta$ $(x,y\in I)$ implies $ax-ay\in \ker \theta$ for all $a\in A$ (S. Veldsman, An overnilpotent radical theory for near-rings. J. Algebra 144 (1991), 248 - 265). One would like to replace the second requirement with some internal condition on the near-ring $N$ only.

What the situation is in the variety of all near-rings is still open. For some information, see S. Veldsman, On ideals and extensions of near-rings. Publ. Math. (Debrecen) 41 (1992), 13 - 22.

(2) Filial rings have been studied extensively and their structure is well-known (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 over-rings. Publ. Math. (Debrecen) 35 (1988), 274 - 279). Characterize the filial near-rings. A partial result can be found in S. Veldsman, On ideals and extensions of near-rings. Publ. Math. (Debrecen) 41 (1992), 13 - 22.

(3) Is a simple zero-symmetric near-ring with identity equiprime? A near-ring $N$ 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 zero-symmetric near-rings? Recall, a class $\mathcal{S}$ of rings (or omega-groups) 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 omega-groups, 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 $(P).$