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Journals / Turkish Journal of Mathematics / 1998 / Cilt: 22 - Sayı: 4

On the Differential Prime Radical of a Differential Ring

D. KHADJIEV
F. ÇALLIALP
Pages
355–368
Publication Date
2022-11-05
DOI
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AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Then every radical {\cal D}-ideal of K is an intersection of {\cal D}-prime {\cal D}-ideals.

📌 3 Key Takeaways

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Objective & Hypothesis: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.
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Key Findings & Data: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.
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Conclusion & Impact:Then every radical {\cal D}-ideal of K is an intersection of {\cal D}-prime {\cal D}-ideals.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Turk. J. Math. · (1998) · 355-368. Full text: pdf Other articles published in the same issue: Turk. J. Math.

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 d is a differentiation of a {\cal Q}-algebra R with 1 then d (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.

Özet

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 d is a differentiation of a {\cal Q}-algebra R with 1 then d (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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Cite This Article

KHADJIEV, D., ÇALLIALP, F. (2022). On the Differential Prime Radical of a Differential Ring. Turkish Journal of Mathematics, Cilt: 22 - Sayı: 4, 355–368. https://ajindex.com/articles/608998

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