ZY环
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摘要
给出ZY环的定义,研究ZY环的一些性质,主要证明了如下结果:设R为ZY环且a∈R,1)若a2=0,则aRaRa=0;2)若a∈aRa,则存在c∈R,使得a=ca2;3)对任意e∈E(R),总有(1-e)aeR(1-e)ae=0;4)设M是R的极大左理想,则对任意e∈E(R),或者e∈M,或者1-e∈M;5)R为nil-semicommutative环;6)设e∈E(R),若ReR=R,则e=1.
In this paper,the author introduces the notion of ZY rings and discusses some properties of ZY rings.The following results for a ZY ring Rare mainly shown:1)If a2=0,then aRaRa=0;2)If a∈aRa,then exists c∈Rsuch that a=ca2;3)If e∈E(R),then(1-e)aeR(1-e)ae=0;4)Let Mbe a maximal left ideal of Rand e∈E(R).Then either e∈Mor 1-e∈M;5)Ris a nil-semicommutative ring;6)If e∈E(R)satisfies ReR=R,then e=1.
In this paper,the author introduces the notion of ZY rings and discusses some properties of ZY rings.The following results for a ZY ring Rare mainly shown:1)If a2=0,then aRaRa=0;2)If a∈aRa,then exists c∈Rsuch that a=ca2;3)If e∈E(R),then(1-e)aeR(1-e)ae=0;4)Let Mbe a maximal left ideal of Rand e∈E(R).Then either e∈Mor 1-e∈M;5)Ris a nil-semicommutative ring;6)If e∈E(R)satisfies ReR=R,then e=1.
引文
[1]WANG L,WEI J C.Central semicommutative rings[J].Indian J Pure Appl Math,2014,45(1):13-25.
[2]WANG Long,WEI Junchao.Weakly semicommutative rings and strongly regular rings[J].Kyungpook Math J,2014,54(1):65-72.
[3]BURCU U,SAIT H,HANDAN K,et al.Rings in which every nilpotent is central[J].Algebras Groups Geom,2013,30(1):1-18.
[4]熊丽丽,李男杰,魏俊潮.CN-环[J].扬州大学学报:自然科学版,2011,14(2):7-9.
[5]WEI Junchao.Some notes on CN rings[J].Bull Malays Math Sci Soc,2014,37(3):25-37.
[6]NICHOLSON W K,YOUSIFF M F.On a theorem of Camillo[J].Commun Algebra,1995,23(14):5309-5314.
[7]REGE M B.On von Neumann regular rings and SF-rings[J].Math Japon,1986,31(6):927-936.
[8]CHEN Weixing.On nil-semicommutative rings[J].Thai J Math,2011,9(1):39-47.
[9]QU Yinchun,WEI Junchao.Some notes on nil-semicommutative rings[J].Turk J Math,2014,38(2):212-224.
[10]HWANG S U,JEON Y C,PARK K S.On NCI rings[J].Bull Korean Math Soc,2007,44(2):215-224.
[2]WANG Long,WEI Junchao.Weakly semicommutative rings and strongly regular rings[J].Kyungpook Math J,2014,54(1):65-72.
[3]BURCU U,SAIT H,HANDAN K,et al.Rings in which every nilpotent is central[J].Algebras Groups Geom,2013,30(1):1-18.
[4]熊丽丽,李男杰,魏俊潮.CN-环[J].扬州大学学报:自然科学版,2011,14(2):7-9.
[5]WEI Junchao.Some notes on CN rings[J].Bull Malays Math Sci Soc,2014,37(3):25-37.
[6]NICHOLSON W K,YOUSIFF M F.On a theorem of Camillo[J].Commun Algebra,1995,23(14):5309-5314.
[7]REGE M B.On von Neumann regular rings and SF-rings[J].Math Japon,1986,31(6):927-936.
[8]CHEN Weixing.On nil-semicommutative rings[J].Thai J Math,2011,9(1):39-47.
[9]QU Yinchun,WEI Junchao.Some notes on nil-semicommutative rings[J].Turk J Math,2014,38(2):212-224.
[10]HWANG S U,JEON Y C,PARK K S.On NCI rings[J].Bull Korean Math Soc,2007,44(2):215-224.