** Authors:**
DJAVVAT KHADJIEV, FETHİ ÇALLIALP

** 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.

** Keywords: **

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