Matrix near-rings and 0-primitivity.
As for rings, the relationships between the radicals of a near-ring and those of the corresponding matrix near-rings have been thoroughly investigated. An intriguing remaining open question is whether the underlying base near-ring of a 0-primitive matrix near-ring must also be \(0\)-primitive. It is already known that the matrix near-ring over a \(0\)-primitive near-ring must be \(0\)-primitive and if the base near-ring is finite, the converse also holds. The authors do not answer the open question here, but they show that it has a positive answer under suitable finiteness conditions: If the \(0\)-primitive matrix near-ring \(\mathbb M_n(R)\) has the dcc on \(\mathbb M_n(R)\)-subgroups, then the base near-ring \(R\) has the dcc on \(R\)-subgroups and is also \(0\)-primitive. Any optimisism of a positive answer in general is slightly dampened by an example provided in the last section where a \(0\)-symmetric generalized matrix near-ring \(\mathbb M_2(R,G)\) is given over a base near-ring \(R\) which is not \(0\)-primitive.
scientific article; zbMATH DE number 6018271