One-sided duo property on nilpotents

Key Findings & Data:We prove the following for a one-sided nilpotent-duo ring $R$: (i) The set of nilpotents in $R$ forms a subring; (ii) Köthe's conjecture holds for $R$; (iii) the subring generated by the identity and the set of nilpotents in $R$ is a one-sided duo ring; (iv) if the polynomial ring $R[x]$ over $R$ is one-sided nilpotent-duo then the set of nilpotents in $R$ forms a commutative ring, and $R[x]$ is an NI ring.

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Conclusion & Impact:Especially we investigate several kinds of conditions under which one-sided nilpotent-duo rings are NI.

🔬Design:Research Article

📍Scope:Comprehensive Sample

⚙️Variables:right (left) nilpotent-duo ring · nilpotent · right (left) duo ring

Abstract

We study the structure of nilpotents in relation with a ring property that is near to one-sided duo rings. Such a property is said to be one-sided nilpotent-duo. We prove the following for a one-sided nilpotent-duo ring $R$: (i) The set of nilpotents in $R$ forms a subring; (ii) Köthe's conjecture holds for $R$; (iii) the subring generated by the identity and the set of nilpotents in $R$ is a one-sided duo ring; (iv) if the polynomial ring $R[x]$ over $R$ is one-sided nilpotent-duo then the set of nilpotents in $R$ forms a commutative ring, and $R[x]$ is an NI ring. Several connections between one-sided nilpotent-duo and one-sided duo are given. The structure of one-sided nilpotent-duo rings is also studied in various situations in ring theory. Especially we investigate several kinds of conditions under which one-sided nilpotent-duo rings are NI.

Keywords

right (left) nilpotent-duo ring nilpotent right (left) duo ring radical NI ring matrix ring ascending chain condition on left (right) annihilators

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Cite This Article

Hong, C. Y., Kim, H. K., Kim, N. K., Kwak, T. K., Lee, Y. (2020). One-sided duo property on nilpotents. Hacettepe Journal of Mathematics and Statistics, Volume: 49 Issue: 6. https://doi.org/10.15672/hujms.571016

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