Quasi-abel环的一些刻画
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摘要
指出quasi-abel环是局部环与Abel环的真正推广,主要研究了quasi-abel环的一些性质:(i)若R为quasi-abel环且f∈E(R),则fRf为quasi-abel环;(ii)R为quasi-abel环当且仅当UTM_2(R)或U_3(R)或R[i,j,k]或SU_3(R)为quasi-abel环;(iii)设R为quasi-abel环,a∈R为正则元,则a∈R~#;(iv)若R为quasi-abel环,则R为直接有限环;(v)设R模J(R)可幂等提升,则R是quasi-abel环当且仅当R/J(R)是Abel环.
Quasi-abel rings are a generalization of local rings and abel rings. Some characterizations of quasi-abel rings are given:(i) If R is a quasi-abel ring and f∈E(R), then fRf is a quasi-abel ring;(ii) R is a quasi-abel ring if and only if UTM_2(R) or U_3(R) or R[i,j,k] or SU_3(R) is a quasi-abel ring;(iii) If R is a quasi-abel ring and a∈R is a regular element, then a∈R~#;(iv) If R is a quasi-abel ring, then R is a directly finite ring;(v) If J(R) is an R module and can be raised idempotent, then R is a quasi-abel ring if and only if R/J(R) is an Abel ring.
Quasi-abel rings are a generalization of local rings and abel rings. Some characterizations of quasi-abel rings are given:(i) If R is a quasi-abel ring and f∈E(R), then fRf is a quasi-abel ring;(ii) R is a quasi-abel ring if and only if UTM_2(R) or U_3(R) or R[i,j,k] or SU_3(R) is a quasi-abel ring;(iii) If R is a quasi-abel ring and a∈R is a regular element, then a∈R~#;(iv) If R is a quasi-abel ring, then R is a directly finite ring;(v) If J(R) is an R module and can be raised idempotent, then R is a quasi-abel ring if and only if R/J(R) is an Abel ring.
引文
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[3] Wei J C,Li L B.Quasi-normal rings[J].Comm Algebra,2010,38(5):1855-1868.
[4] 周慧敏,杜昕彦,魏俊潮.Abel环的一些刻画[J].扬州大学学报(自然科学版),2015,18(4):16-18.
[5] 赵旭东,王姗,魏俊潮.直接有限环的一些刻画[J].大学数学,2017,33(6):7-11.
[6] 马丽,史丽妍,魏俊潮.群可逆元的一些刻画[J].大学数学,2018,34(2):1-6.
[7] Bu C J,Zhang K Z.Group inverse for the block matrices with an invertible Subblock[J].Appl Math Comput,2009,215(1):132-139.
[8] Hartwig R E.Block generalized inverses[J].Arch.Retional Mech.Anal,1976,61(3):197-251.
[9] Koliha J J,Patricio P.Elements of rings with equal spectral idempotents[J].J.Austr.Math.Soc.,2002,72(1):137-152.
[10] Yao H,Wei J C.EP elements and *-strongly regular rings[J].Filomat,2018,32(1):117-125.
[11] 魏俊潮.直接有限环[J].扬州大学学报(自然科学版),2005,8(2):1-3.