Abstract
By a near-ring we mean a right near-ring.J0r, the right Jacobson radical of type 0, was introduced for near-rings by the first and second authors. In this paper properties of the radicalJ0rare studied. It is shown thatJ0ris a Kurosh-Amitsur radical (KA-radical) in the variety of all near-ringsR, in which the constant partRcofRis an ideal ofR. So unlike the left Jacobson radicals of types 0 and 1 of near-rings,J0ris a KA-radical in the class of all zero-symmetric near-rings.J0ris nots-hereditary and hence not an ideal-hereditary radical in the class of all zero-symmetric near-rings.
Highlights
R denotes a right near-ring and all near-rings considered are right near-rings and not necessarily zero-symmetric.In 1, 2, the first author studied the structure of near-rings in terms of right ideals, and showed that as in rings, matrix units determined by right ideals identify matrix near-rings
In 6, Wedderburn-Artin theorem was extended to near-rings, and some generalizations of it were presented
It is known that the left Jacobson radicals of types 0 and 1 are not KA-radicals in the class of all International Journal of Mathematics and Mathematical Sciences zero-symmetric near-rings, and only the left Jacobson radicals of types 2 and 3 are KA-radicals in the class of all zero-symmetric near-rings
Summary
R denotes a right near-ring and all near-rings considered are right near-rings and not necessarily zero-symmetric.In 1, 2 , the first author studied the structure of near-rings in terms of right ideals, and showed that as in rings, matrix units determined by right ideals identify matrix near-rings. J0r, the right Jacobson radical of type 0, is a KA-radical in the class of all zero-symmetric near-rings. J0r is not s-hereditary, and not an ideal-hereditary radical in the class of all zero-symmetric near-rings. F denotes the class of near-rings R, in which the constant part Rc of R is an ideal of R.
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