Special Right Jacobson Radicals for Right Near-rings
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In this paper three more right Jacobson-type radicals, Jrgν, are introduced for near-rings which generalize the Jacobson radical ofrings, ν ∈ {0, 1, 2}. It is proved that Jrgν is a special radical in the class of all near-rings. Unlike the known right Jacobson semisimple near-rings, a Jrgν-semisimplenear-ring R with DCC on right ideals is a direct sum of minimal right ideals which are right R-groups of type-gν, ν ∈ {0, 1, 2}. Moreover, a finite right g2-primitive near-ring R with eRe a non-ring is a near-ring of matrices over a near-field(which is isomorphic to eRe), where e is a right g2-primitive idempotent in R.
Abstract
In this paper three more right Jacobson-type radicals, Jrgν, are introduced for near-rings which generalize the Jacobson radical ofrings, ν ∈ {0, 1, 2}. It is proved that Jrgν is a special radical in the class of all near-rings. Unlike the known right Jacobson semisimple near-rings, a Jrgν-semisimplenear-ring R with DCC on right ideals is a direct sum of minimal right ideals which are right R-groups of type-gν, ν ∈ {0, 1, 2}. Moreover, a finite right g2-primitive near-ring R with eRe a non-ring is a near-ring of matrices over a near-field(which is isomorphic to eRe), where e is a right g2-primitive idempotent in R.
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