On para-special chains of near-rings (Q1105672)

A (left) near-ring N is said to have the property a, if, for any \(a\in N\), the set \([0]_ a=\{z\in N|\) \(za=0a\}\) is an ideal and an N- subgroup. If N is a near-ring, and S,S'\(\in {\mathbb{P}}(N)\), put \(A_ d(S)=\{x\in N|\) \(Sx=0\}\), \(A_ s(S')=\{x\in N|\) \(xS'=0\}\), \(A(S,S')=\{x\in N|\) \(SxS'=0\}\). Put \(A_ 0(N)=\{A_ d(S)|\) \(S\subseteq N\), S finite\(\}\), \(A'(N)=\{A_ s(S')|\) S'\(\subseteq N\), S' finite\(\}\), \(A''(N)=\{A(S,S')|\) S,S'\(\subseteq N\), S and S' finite\(\}\), \(A_ 1(N)=A_ 0(N)\cup A'(N)\cup A''(N).\) The author demonstrates that, if N is finite, zero-symmetric, with the property a, and such that the quotient of N with respect to the nil radical is integral, then N has a chain of ideals of \(A_ 1(N)\), \(0=T_ 0\subset T_ 1\subset...\subset T_ m=N\), where 1) every quotient is without proper right annihilators; 2) for \(i=1,2,...,m-3\), the set of the right annihilators of \(T_{i+2}/T_ i\) is a subset of \(\{T_{i+1}/T_ i,T_{i+2}/T_ i\}\); 3) the set of the right annihilators of \(T_ m/T_{m-2}\) is contained in \(\{T_{m-2}/T_{m-2},T_{m-1}/T_{m- 2},T_ m/T_{m-2}\}\). Call a chain of ideals N, having the aforesaid properties 1), 2), 3) a para-special chain. A necessary and sufficient condition, so that a finite, zero-symmetric near-ring, with the property a, has a para-special chain, is that, N be nilpotent, or the quotient of N with respect to its nil radical be integral.