In this paper we show first that the sum of the nil right ideals of a distributive near-ring N and the sum of the nil left ideals of N have the same two-sided annihilator. This ideal-here denoted by α(N)-proves useful in studying the existence of nil non-nilpotent ideals when the near-ring is weakly semiprime. In particular, we show that N has a nil non-nilpotent left ideal if and only if α(N) is properly contained in N and that every one-sided non-nilpotent ideal contains a nil non-nilpotent one-sided ideal if and only if α(N) coincides with the (two-sided) annihilator of N. Finally we prove that N has a two-sided nil non-nilpotent ideal if and only if the ring N/α(N) has one.
On the existence of nil ideals in distributive near-rings / S. De Stefano, S. Di Sieno. - 137:C(1987), pp. 53-58. ((Intervento presentato al convegno 9th International Conference on Near-rings and Near-fields tenutosi a Oberwolfbach nel 1987 [10.1016/S0304-0208(08)72286-2].
On the existence of nil ideals in distributive near-rings
S. De StefanoPrimo
;S. Di SienoUltimo
1987
Abstract
In this paper we show first that the sum of the nil right ideals of a distributive near-ring N and the sum of the nil left ideals of N have the same two-sided annihilator. This ideal-here denoted by α(N)-proves useful in studying the existence of nil non-nilpotent ideals when the near-ring is weakly semiprime. In particular, we show that N has a nil non-nilpotent left ideal if and only if α(N) is properly contained in N and that every one-sided non-nilpotent ideal contains a nil non-nilpotent one-sided ideal if and only if α(N) coincides with the (two-sided) annihilator of N. Finally we prove that N has a two-sided nil non-nilpotent ideal if and only if the ring N/α(N) has one.
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