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| The Jacobson Relational Radical and the Jacobson Radical |
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Citation： |
Li Lide.The Jacobson Relational Radical and the Jacobson Radical[J].Chinese Annals of Mathematics B,1982,3(6):745~752 |
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Net amount： 889 |
Authors： |
Li Lide; |
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| Abstract： |
In this paper it is shown that the intersection of all w-relations of a hemiring R is exactly the intersection of all primitive relations of this hemiring, and is called J-relational radical of the himiring R. Several properties of the J-relational radical are described. The radical R of a hemiring R is defined by the set {x\in R|x\tau 0}, where \tau is the J-relational radical of R. We obtain independently following results: 1.R is a right quasi-regular ideal which contains every right quasi-regular right ideal. 2.R=\cap {(0:M)|M,, irreducible cancellative right R-semimodule. The term "Jacobson semisimple” in ring theory is generalized to hemirings by defining “J-relational semisimple.” It is proved that if \tau is the Jacobson relational radical of a hemiring R, then R/\tau is J-relational semisimple. Finally the structure theorem of the hemirings is given. A hemiring is J-relational simisimple if and only if it is isomorphic to a subdirect sum of completely primitive hemirings. |
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