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UBC Theses and Dissertations

Radicals in near-rings Thompson, Charles Jeffrey James

Abstract

An algebraic system which satisfies all the ring axioms with the possible exceptions of commutativity of addition and the right distributive law is called a near-ring. This thesis is intended as a survey of radicals in near-rings, and an organization of the theory which has been developed to date. Because of the absence of the right distributive law, the zero element of a near-ring need not annihilate the near-ring from the left. If we impose the condition that 0 • p = 0 for all elements p of a near-ring P, then we call P a C-ring. This condition is ensured if we demand that the near-ring P be generated, as an additive group, by a set S of elements of P such that (P₁+ P₂)s = P₁s + P₂S for all P₁, P₂ in P, and s in S. In this case, P is said to be distributively generated by S. The work is divided into three main sections; the first deals with general near-rings, the second with C-rings, and the third with distributively generated near-rings. Appendix I gives a proof of a vital result for distributively generated near-rings, due to Laxton [11]; appendix II introduces a little used radical due to Deskins [6]; appendix III is included as a concrete example of a near-ring and its theory, due to Berman and Silverman [2].

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