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When nilpotents are contained in Jacobson radicals

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대한수학회 (The Korean Mathematical Society)|2018.09|대한수학회지 (Journal of the KMS)|제55권 제5호|pp. 1193-1205

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참고문헌

키워드

Jacobson radicalNI ringnilradicalNJ ringKothe's conjectureright quasi-duo ring(skew) polynomial ring
주제분류수학

초록

We focus our attention on a ring property that nilpotents are contained in the Jacobson radical. This property is satisfied by NI and left (right) quasi-duo rings. A ring is said to be $\it NJ$ if it satisfies such property. We prove the following: (i) Kothe's conjecture holds if and only if the polynomial ring over an NI ring is NJ; (ii) If $R$ is an NJ ring, then $R$ is exchange if and only if it is clean; and (iii) A ring $R$ is NJ if and only if so is every (one-sided) corner ring of $R$.

Abstract

We focus our attention on a ring property that nilpotents are contained in the Jacobson radical. This property is satisfied by NI and left (right) quasi-duo rings. A ring is said to be $\it NJ$ if it satisfies such property. We prove the following: (i) Kothe's conjecture holds if and only if the polynomial ring over an NI ring is NJ; (ii) If $R$ is an NJ ring, then $R$ is exchange if and only if it is clean; and (iii) A ring $R$ is NJ if and only if so is every (one-sided) corner ring of $R$.

참고문헌 (33)

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