Weakly regular rings with ACC on annihilators and maximality of strongly prime ideals of weakly regular rings |
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Authors: | Chan Yong Hong Kyoung Hwan Kim Yang Lee |
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Institution: | a Department of Mathematics and Research Institute for Basic Sciences, Kyung Hee University, Seoul 130-701, Republic of Korea b Department of Mathematics Education, Pusan National University, Pusan 609-735, Republic of Korea c College of Liberal Arts, Hanbat National University, Daejeon 305-719, Republic of Korea |
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Abstract: | Ramamurthi proved that weak regularity is equivalent to regularity and biregularity for left Artinian rings. We observe this result under a generalized condition. For a ring R satisfying the ACC on right annihilators, we actually prove that if R is left weakly regular then R is biregular, and that R is left weakly regular if and only if R is a direct sum of a finite number of simple rings. Next we study maximality of strongly prime ideals, showing that a reduced ring R is weakly regular if and only if R is left weakly regular if and only if R is left weakly π-regular if and only if every strongly prime ideal of R is maximal. |
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Keywords: | 16D25 16E50 16P60 |
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