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## Turkish Journal of Mathematics

#### Article Title

On the Differential Prime Radical of a Differential Ring

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#### Abstract

In this paper we have obtained the following results for a differential ring (associative or nonassociative): (1) For a differential ring ({\cal D}-ring) we introduce definitions of a {\cal D}-prime {\cal D}-ideal, {\cal D}-semiprime {\cal D}-ideal and a strongly {\cal D}-nilpotent element. We define the {\cal D}-prime radical as the intersection of all {\cal D}-prime {\cal D}-ideals. For any {\cal D}-ring the {\cal D}-prime radical, the intersection of all {\cal D}-semiprime {\cal D}-semiprime {\cal D}-ideals and the set of all strongly {\cal D}-nilpotent elements are equal. (2) For a {\cal D}-ring we introduce a definition of an s-nilpotent {\cal D}-ideal. If a {\cal D}-ring satisfies the ascending chain condition for {\cal D}-ideals then its {\cal D}-prime radical is s-nilpotent. (3) Let {\cal Q} be a field of rational numbers. If \delta is a differentiation of a {\cal Q}-algebra R with 1 then \delta (Pr.rad(R))\subseteq Pr. rad(R). (4) Let K be a differential ring. Then every radical {\cal D}-ideal of K is an intersection of {\cal D}-prime {\cal D}-ideals.

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