1、关于 clean 环的几个注记(?p?)C?C?C?C?C?C?g44?2008?g6?7C?JournalofNorthwestNormalUniversity(NaturalScience)C?C?C?C?C?C?Vol144C?2008C?No16C?p?k?7:20071120;?F?k?p?C?7:20080516?0?:?1?S?(19821e),?,?i?,?e?7.?e?t?K?8?.zjy1clean?O?1?S?(?U?J?4?j?l?,?i?C?730070)?C?:?l?clean?M?Gclean?;?e,clean?u?y?f?Gclean?.?H?:clean?;
2、?v?-;?clean?;?y?f?Z?8?:O15313C?C?C?C?H?6?(:AC?C?C?C?m?:10012988;?(2008)0620024202SeveralnotesforcleanringsZHANGJunyu(CollegeofMathematicsandInformationScience,NorthwestNormalUniversity,Lanzhou730070,Gansu,China)Abstract:Itisprovedthatquasicleanringsareexactlycleanrings.Furthermore,accordingtotheexis
3、tingconclusionthatpolynomialringsareneverclean,thepaperpresentsseveralspecialexamplesforfactorringsofpolynomialringstobecleanrings.Keywords:cleanring;quasiregularelement;quasicleanring;polynomialringC?C?1977?,Nicholson?clean?,?B?l?y?0?e?clean?,?:?Kclean?,?G?clean?,?Dclean?,nclean?,?clean?1?e?.?,?H1?
4、clean?,?j?Y?.?g?l:?clean?M?Gclean?;?e,clean?R?u?y?f?Rx?Rx/(x2)?Gclean?,clean?R?u?y?-?y?f?Rx1,5?,xk?Rx1,5?,xk/(x21,5?,x2k)?Gclean?.?g?Z?R?P?6?o?W?-?J?,U(R)?KR?W?T?N?,I(R)?KR?+?-?N?J,J(R)?KR?Jacobson?F,Tn(R)?KR?u?u?o?b?.?-?1a5JR?Gclean?-2,?a=e+u,?Ze5JI(R),u5JU(R).?R?Gclean?,?R?Z?O?-?1?Gclean?-.1C?clea
5、n?l?clean?-?1a5JR?G?v?-,?a1d5JR,?a1d+a+a1da=0,?(1+a1d)(1+a)=1.?a1d5JR,?a+a1d+aa1d=0,?(1+a)(1+a1d)=1,?v?a?K?v?-.?a?G?v?-?G?v?-,?v?a?K?v?-3.?R?Z?-?1a?G?clean?-,?a=e+q,?Ze5JI(R),q?GR?Z?v?-.?R?G?clean?,?R?Z?O?-?1?G?clean?-1.?81C?R?Gclean?e?R?G?clean?.?lC?R?Gclean?,?v?fx5JR,?e5JI(R),u5JU(R),?x+1=e+u,?T?x
6、=e+(u-1).?K1+(u-1)=u?G?8?-,?T?u-1?GR?42Email:hangunu999sinacom2008?g6?7C?C?1?S?:?clean?O?C?2008C?No16C?C?C?SeveralnotesforcleanringsC?v?-.?R?G?clean?.?,?R?G?clean?,?v?fx5JR,?e5JI(R)?R?v?-q,?x-1=e+q,?T?x=e+(1+q).?n1+q5JU(R)(?M?v?-?),?R?Gclean?.C?C?Ct?81?l,?H1?clean?y?u?G?clean?l.?H1?Z?219?l?R?G?,?v?y
7、?f?Rx?a?G?clean?.?H4?Z,Nicholson?Zhou?l?:?f?R,?y?f?Rx?a?Gclean?.?T?M?81?:?1C?R?G?f?,?v?y?f?Rx?a?G?clean?.?R?G?O?,I?GR?8?,?R/I?G?+?-?P?,?a2-a5JI,?v?e5JI(R),?e-a5JI.?e?Pa.?F?H5,?R?Gexchange?e?R?f?(?m?)?8?A,R/A?G?+?-?P?.?H1?Z?2111?l?:?R?Gexchange?,A?GR?8?eAJ(R),?vR?G?clean?e?R/A?G?clean?.?u,R?Gexchange
8、?M?,?t?o:?1C?A?G?R?8?eAJ(R),?vR?G(?)clean?e?R/A?G(?)clean?eR/A?G?+?-?P?.?lC?H6?6.C?C?Ct2C?clean?u?y?f?clean?R?G?O?,V?G?O?g?(?a?o?W?-)?eV?G(R,R),?:?fr5JR,?fv,w5JR,?o(vw)r=v(wr),(vr)w=v(rw),(rv)w=r(vw)?N?.?Abel?R?V?u?:(r,v)(s,w)=(rs,rw+vs+vw),?v?w?R?V?l?N?O?o?W?-(1,0)?,?KI(R;V),?B?KR?y?V?8?1.?A?C,?S=R
9、?A,?ZR?GS?,A?GS?8?,?vS?I(R;V)4.?s?81C?r?,?,?8?1S=R?V?Gclean?:()R?G?;(2)?f?v5JV,?w5JV,?v+w+vw=0.?lC?8?H4?7?Z?G?j?l?.C?C?Ct?s?82C?R?Gclean?,S=aij5JTn(R)a11=5?=ann,?vS?Gclean?.?lC?R1d=aij5JTn(R)a11=5?=ann,aij=0,i5?j.?/?pR?R1d.?V=aij5JTn(R)a11=5?=ann=0,?v?o?b?l?,?w?Y?U?fR1d?GS?,V?GS?8?,?eS=R1d?V?I(R;V).
10、?,R?Gclean?.?;,?v5JV,?w?det(1R1d+v)=1,?1R1d+v?GS?8?b,?l(1,v)?GI(R;V)?Z?8?-,?(a,w)5JI(R;V),?(1,0)=(1,v)(a,w),?v+w+vw=0.?F?s?81,S?Gclean?.C?C?Ctclean?p?Gclean?,?R?Gclean?,?y?f?Rx?a?Gclean?.?r?l?q?y?f?o?clean?.?2C?R?Gclean?,?vRx/(x2)?Gclean?.?lC?KRx/(x2)?aC?b0C?aa,b5JR,?T?J?s?82?.C?C?Ct?u?C?y?-?y?f?.?1
11、C?R?G?f?,k?G?f?,x1,5?,xk?G?O?-,?v?y?-?y?f?Rx1,5?,xk?a?Gclean?.?lC?Rx1,5?,xk=Rx1,5?,xk-1xk,?J?1,?.C?C?Ct?3C?R?Gclean?,k?G?f?,x1,5?,xk?G?O?-,?vRx1,5?,xk/(x12,5?,xk2)?Gclean?.?lC?h:(Rx1,5?,xk-1/(x12,5?,xk-12)xkC?C?Rx1,5?,xk/(x12,5?,xk2),5tni=0fixki5ti=0,1fixki+(x12,5?,xk2),?Z,fi5JRx1,5?,xk-1/(x12,5?,xk
12、-12).?w?h?G?n?(.(?r?g?)521clean28(?p?)C?C?g44?C?JournalofNorthwestNormalUniversity(NaturalScience)C?C?Vol144C?C?u2iv2i+1?u2i+1v2i(i=1,2,5?).?m,?t?T?Z?t?C?,?,?F?W?,?u1vi?lunvj(i,j=1,2,5?,n?ei5?j),?2?m3?Y?,?q?,?Pn5Pn?Z?n(n-3)?2?,?x?r,?o?K4?G2.?o4?s1d(Pn5Pn)n2-3n+9.?l?G:?n5?264?,?G?i?H2?Z?g?,?n?l?r?Z?w
13、?Y?.C?C?Ct?1C?mn5?4?,m+25?4?s1d(Pm5Pn)mn-3n+9.?lC?3?82,?a?Pm?u?on?O?,?3?u?Pn5Pn?Y?7?,?B?l?(m-n)n?2?1?,?q?mn-3n+9?,?e?rPm5Pn?Z?f?K4?a?G2,?J?s?81,?om+25?4?s1d(Pm5Pn)mn-3n+9.C?C?Ct?4?H:1C?FERTING,RASPAUDA,REEDB.StarcoloringofgraphsJ.JGraphTheory,2004,47:163183.2C?i?7,?C?.?j?a?f?7?7?O?u?YJ.?j?:?p?,2008,
14、44(2):9899.3C?BONDYA,MURTYUSR.GraphTheorywithApplicationsM.NewYork:Macmillan,LongdonandElsevier,1976.4C?GRNBAUMB.AcycliccolouringsofplanargraphsJ.IsrealJMath,1973,14:390408.5C?.Pm5Pn?K?A?J.?U?J?4?j?:?p?,2005,41(1):1315.6C?BAZAGANC,HARKATBENHAMDINEA,LIH,etal.Onthevertexdistinguishingproperedgecolorin
15、gofgraphJ.JofCombineTheory(SerB),1999,75:288301.7C?ZHANGZhongfu,LIULingzhong,WANGJianfang.AdjacentstrongedgecoloringofgraphsJ.AppliedMathematicsLetters,2002,15:623626.8C?1?,?,?,?.?a?j?4?f?,?K?A?J.?,2006,49(3):703708.(?9?m?C?k)(?u?g25?)C?C?r?Ker(h)=(xk2).?5tni=0fixki5JKer(h),?v5ti=0,1fixki5J(x12,5?,xk2),?5tni=0fixki=5ti=0,1fixki+5tni=2fixki5J(x12,5?,xk2).?T?f0=0,f1=0,?Ker(h)(xk2).?t?/?p?N?,?oKer(h)=(xk2
