拟对偶双边模与广义矩阵环
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摘要
零化子在研究对偶环，拟对偶环及对偶双边模中起着非常重要的作用．在第二章中我们首先定义了左拟对偶双边模．设环R是带有单位元的结合环，M_R是一个右R-酉模，令S=End(M_R)，易知_sM_R是一个(S，R)-双边模．一个双边模_sM_R是一个左拟对偶双边模，如果M_R的每一个本质子模K和S的每一个本质子左理想L分别满足r_Ml_s(K)=K和l_sr_M(L)=L．在第2.1节我们研究了拟对偶双边模的性质．如果_sM_R是一个左拟对偶双边模，则我们有：
(1)r_Ml_s(Soc(M_R))-Soc(M_R)和l_sr_M(Soc(_sS))-Soc(_sS)；
(2)如果M_R是一个cs-模，则Soc(M_R)在M_R中是本质的；
(3)如果M_R是非M-奇异的，则M_R是半单的；
(4)如果M_R在σ[M]中投射并且半单，则M_R是非M-奇异的．
在第2．2节中我们讨论了拟对偶双边模和对偶双边模的关系．我们得到：一个左拟对偶双边模如果满足下列条件之一，则它将成为一个左对偶双边模：
(ⅰ)_sM是单内射的并且M_R是一个M-单内射kasch-模；
(ⅱ)M_R是一个M-单内射kasch-模并且对_sS的任意两个理想，有r_M(L_1∩L_2)=r_M(L_1)+r_M(L_2)；
(ⅲ)_sM是单内射的且对M_R的任意两个子模，有l_s(A∩B)=l_s(A)+l_s(B)．
2 在第2．3节中我们将拟对偶性应用于smash积代数R#H，部分解决了半素问题。即令H是一个有限维半单Hopf代数，R是一个H-模代数，如果R是左拟对偶的且半素的，则R#H是半素的．
在第三章中我们讨论了广义矩阵环．我们首先讨论了广义矩阵环的根，得到了广义矩阵环的稠密性定理和Wedderburn-Artin定理．其次，我们从根的意义上刻画了方向图的性质．最后，我们给出了广义矩阵环A，A_ji-环A_ij和环A的Von Neumann正则根之间的关系．我们得到了以下主要结论：
(1)如果r是环的一个超幂零根，A是一个广义矩阵环，则r(A)是A的一个广义矩阵理想．
p）如果厂是环的一个超幂零根，则厂是一个N根当且仅当对任何广义矩阵环A，
有厂（A） 刀r（An）h，j EI｝
O）如果J有广义矩阵左且右一零因子，则 s州．rn（．）一二《rn（．），’ e＊X
O）令厂－＠。Vg是一个群分级向量空间，其中 Vg是有限维的，
A＃－HOIn（V，，K＊i。G，jEG·A－Z｛A＃卜j。G），Ng
g·m·rn（A）＝Z｛rn（A。）i，jEG｝＝A·
（5） r（A）一g．m．r（A）一r＜A）一Z迁r（A了）i，jEI｝，R中r表示 ＆，r＊＆rl．
Annihilator plays an important role in studying dual-rings, quasi-dual rings and dual-bimodules. In chapter two we first define left quasi-dual bimodules. Assume ring R is associative with identity and module MR is a unitary right R-moaule, let S=End(MR\ it is trivial that $MR is a (S, 7?)-bimodule. A bimodule S&/R is a left quasi-dual bimodule if every essential submodule K of MR and every essential left ideal L of S satisfy with
rMls(K)=K and lsrM(L)=L respectively. At first, we study the properties of
quasi-dual bimodules. If sMR is a left quasi-dual bimodule, we obtain the following conclusions:
(1) rMls(Soc(MR))-Soc(MR) and lsrM(Soc(sSy)-Soc(sS);
(2) If MR is a cs-module, then SOC(MR) is essential in MR,
(3) If MR is non-M-singular, then MR is semisimple;
(4) If MR is projective in o[M] and semisimple, then MR is non-M-singular.
Next, we discuss the relations between left quasi-dual bimodules and left dual-bimodules, we obtain that a left quasi-dual bimodule is a left dual bimodule if it satisfies one of the following conditions:
( i ) sM is minimal injective and MR is a M-minimal injective kasch-module;
( ii ) MR is a M-minimal injective kasch-module and for any two ideals LI and L2 of
SS rM(L1 n L2)-rw(L1)+rM(I2);
(iii) sM is minimal injective and for any two submodules A and B of MR,
Lastly, we applicate the quasi-duality on smash product algebra R#H, and obtain an answer of the semiprime problem, i.e., let H be a finite-dimensional semisimple Hopf algebra and R be an H-module algebra, if R is left quasi-dual and semiprime, then R#H is semiprime.
In chapter three we study generalized matrix rings, we first study the radical of the
generalized matrix ring A, obtained Density Theorem and Wedderburn-Artin Theorem of generalized matrix rings. Next, we characterized some properties of directed graph by means of radical. Finally, we give the relations among the Von Neumann regular radicals of generalized matrix ring A, Aji-ring Ajjand ring A. We obtain the following conclusions.
(1) If r is a super nil radical of a ring, A is a g.m.ring, then r(A) is a g.m.ideal of A.
(2) If r is a super nil radical of a ring. Then r is a N-radical if and only if for any
g.m.ring.4, r(A) -
(3) If A has g.m.left and right no zero divisors then g.m.rn(A) =
(4) Let V = is a group graded vector space, where V& is finitely dimensional.
(5) r(A) = gmr(A) = r(A) = , where r doenotes rb, rk or rL
(1)r_Ml_s(Soc(M_R))-Soc(M_R)和l_sr_M(Soc(_sS))-Soc(_sS)；
(2)如果M_R是一个cs-模，则Soc(M_R)在M_R中是本质的；
(3)如果M_R是非M-奇异的，则M_R是半单的；
(4)如果M_R在σ[M]中投射并且半单，则M_R是非M-奇异的．
在第2．2节中我们讨论了拟对偶双边模和对偶双边模的关系．我们得到：一个左拟对偶双边模如果满足下列条件之一，则它将成为一个左对偶双边模：
(ⅰ)_sM是单内射的并且M_R是一个M-单内射kasch-模；
(ⅱ)M_R是一个M-单内射kasch-模并且对_sS的任意两个理想，有r_M(L_1∩L_2)=r_M(L_1)+r_M(L_2)；
(ⅲ)_sM是单内射的且对M_R的任意两个子模，有l_s(A∩B)=l_s(A)+l_s(B)．
2 在第2．3节中我们将拟对偶性应用于smash积代数R#H，部分解决了半素问题。即令H是一个有限维半单Hopf代数，R是一个H-模代数，如果R是左拟对偶的且半素的，则R#H是半素的．
在第三章中我们讨论了广义矩阵环．我们首先讨论了广义矩阵环的根，得到了广义矩阵环的稠密性定理和Wedderburn-Artin定理．其次，我们从根的意义上刻画了方向图的性质．最后，我们给出了广义矩阵环A，A_ji-环A_ij和环A的Von Neumann正则根之间的关系．我们得到了以下主要结论：
(1)如果r是环的一个超幂零根，A是一个广义矩阵环，则r(A)是A的一个广义矩阵理想．
p）如果厂是环的一个超幂零根，则厂是一个N根当且仅当对任何广义矩阵环A，
有厂（A） 刀r（An）h，j EI｝
O）如果J有广义矩阵左且右一零因子，则 s州．rn（．）一二《rn（．），’ e＊X
O）令厂－＠。Vg是一个群分级向量空间，其中 Vg是有限维的，
A＃－HOIn（V，，K＊i。G，jEG·A－Z｛A＃卜j。G），Ng
g·m·rn（A）＝Z｛rn（A。）i，jEG｝＝A·
（5） r（A）一g．m．r（A）一r＜A）一Z迁r（A了）i，jEI｝，R中r表示 ＆，r＊＆rl．
Annihilator plays an important role in studying dual-rings, quasi-dual rings and dual-bimodules. In chapter two we first define left quasi-dual bimodules. Assume ring R is associative with identity and module MR is a unitary right R-moaule, let S=End(MR\ it is trivial that $MR is a (S, 7?)-bimodule. A bimodule S&/R is a left quasi-dual bimodule if every essential submodule K of MR and every essential left ideal L of S satisfy with
rMls(K)=K and lsrM(L)=L respectively. At first, we study the properties of
quasi-dual bimodules. If sMR is a left quasi-dual bimodule, we obtain the following conclusions:
(1) rMls(Soc(MR))-Soc(MR) and lsrM(Soc(sSy)-Soc(sS);
(2) If MR is a cs-module, then SOC(MR) is essential in MR,
(3) If MR is non-M-singular, then MR is semisimple;
(4) If MR is projective in o[M] and semisimple, then MR is non-M-singular.
Next, we discuss the relations between left quasi-dual bimodules and left dual-bimodules, we obtain that a left quasi-dual bimodule is a left dual bimodule if it satisfies one of the following conditions:
( i ) sM is minimal injective and MR is a M-minimal injective kasch-module;
( ii ) MR is a M-minimal injective kasch-module and for any two ideals LI and L2 of
SS rM(L1 n L2)-rw(L1)+rM(I2);
(iii) sM is minimal injective and for any two submodules A and B of MR,
Lastly, we applicate the quasi-duality on smash product algebra R#H, and obtain an answer of the semiprime problem, i.e., let H be a finite-dimensional semisimple Hopf algebra and R be an H-module algebra, if R is left quasi-dual and semiprime, then R#H is semiprime.
In chapter three we study generalized matrix rings, we first study the radical of the
generalized matrix ring A, obtained Density Theorem and Wedderburn-Artin Theorem of generalized matrix rings. Next, we characterized some properties of directed graph by means of radical. Finally, we give the relations among the Von Neumann regular radicals of generalized matrix ring A, Aji-ring Ajjand ring A. We obtain the following conclusions.
(1) If r is a super nil radical of a ring, A is a g.m.ring, then r(A) is a g.m.ideal of A.
(2) If r is a super nil radical of a ring. Then r is a N-radical if and only if for any
g.m.ring.4, r(A) -
(3) If A has g.m.left and right no zero divisors then g.m.rn(A) =
(4) Let V = is a group graded vector space, where V& is finitely dimensional.
(5) r(A) = gmr(A) = r(A) = , where r doenotes rb, rk or rL
引文
[1] R.Baer. Rings with duals.Amer.J.Math,1943,65:569-584.
[2] M.Hall.A type of algebraic closure.Ann.of Math, 1939,40: 360-369.
[3] I.Kaplansky.Dual rings.Ann.of Math, 1948,49:689-701.
[4] I.A. Skornjakov.More on quasi-Frobenius rings. Mat.Sb, 1973,92: 518-529.
[5] C.R.Hajarnavis and N.C.Norton.On dual rings and their modules. J.Algebra, 1985,93:253-266.
[6] S.S.Page and Yiqiang Zhou. Quasi-dual rings.Comm.Algebra,2000,28(1): 489-504.
[7] M.E.Sweedler. Hopf Algebras.Benjamin. New York: Benjamin press, 1969.
[8] Shouchuan Zhang. The radicals of Hopf module algebras. Chinese Ann. Mathematics, Ser B, 1997,18(4): 495-502.
[9] 陈维新，张寿传．广义矩阵环的特殊根．浙江大学学报，1995，29(6)：639-646．
[10] 张寿传和陈维新．Г-环的Baer根和根的一般理论．浙江大学学报(自科版)，1991，25：719-724．
[11] W.E.Coppage and J.Luh. Radicals of gamma rings.J.Maath.soc.Japan, 1971,23(1):40-52.
[12] Weixin Chen and shouchuan Zhang. The module theoretic characteriation of special and supernilpotent radicals for Г -rings. Math. Japonica, 1993,38(3): 541-547.
[13] Booth G L. Supernilpotent radical of Г-rings. Acta Math, Hungarica, 1989, 54(3-4):201-208
[14] S.Kyuno.Nobusawa's gammar rings with right and left unities.Math.Japonice, 1980,25: 180-190.
[15] 刘绍学．加法范畴的Baer根和Levitzki根．北京师大学报，1987，4：13-27．
[16] G.L.Booth. On the radicals of Г_N-rings. Math. Japonica, 1987,32(3): 357-372.
[17] Shoaxue Liu.The geometric properties of directed graph and the algebraic poperties of path algebra. Acta mathematica sinica, 1988,31(4): 433-487.
[18] Shouchuan Zhang. The Baer radical of generalized matrix rings. In:Proceedings of the sixth SIAM conference on parallel processing for scientific computing. American, 1993.
[19] 陈维新．Г-环的最大Von Neumann正则理想．浙江大学学报，1981，3:68-72．
[20] Frank W.Anderson and Kent R.Fuller.Rings and Categories of Modules.Springer GTM, 1973.
[21] Nathan Jacobson.Basic aigebra Ⅱ. San Franciso: W.H.Freeman and Company, 1974.
[22] R.Wibauer.Foundations of Module and Ring Theory.Gordon and Breach: Reading, 1991.
[23] Nguyen Viet Dung, Dinh Van Huynh and Patrick F.Smith,etc. Extending Modules.Heinrich Heine University Dsseldorf, 1994.
[24] A.O.AI-attas and N.Vanaja. On ∑ -extending modules.Comm. algebra, 1997,25(8): 2365-2393.
[25] R.Wisbauer. Generalized co-semisimple modules.Comm.Algebra, 1990, 18:4235-4253.
[26] Shenglin Zhu.On rings over which every flat left modules is finitly projective. J.Algebra,1991,139: 311-321.
[27] F.A.Szasz.Radicals of rings.New York:Jahn wiley and Sons, 1982.
[28] Y.Krata and K.Koike. Dual-bimodules and finitely cogenerated modules Math.J.Okayama Univ, 1995, 37:91-98.
[29] W.K.Nicholson and J.K.Park. Principally Quasi-injective Modules. Comm. Algebra, 1999,27(4): 1683-1693.
[30] W.K.Nicholson and M.F.Yoush, Principally injective rings, J.Algebra 174, 77-93(1995).
[31] S.Montgomery. Hopf algebras and their actions on rings, CBMS Number 82, Published by AMS, 1993.
[32] R.J.Blattner, M.Cohen and S.Montgomery. Crossed products and inner actions of Hopf algebras. Transactions of the AMS. 1986,298(2):671-711.
[33] S.Montgomery. Crossed products of Hopf algebras and enveloping algebras, in Perspectives in Ring Theory. Kluwer 1988, 253-268.
[34] Shouchuan Zhang. Braided Hopf algebra. Chang Sha: Hunan Normal University press, 1999.
[35] Miriam Cohen and Davida Fishman. Hopf algebra actions. J.of Algebra, 1986, 100:363-379.
[36] 陈维新和张寿传．有单位元的单Г-环类的根．数学研究，1996，16(2)：263-268．．
[37] A.D.Sands. Radicals and Morita Contexts. J.Algebras, 1973, 24: 335-345.
[38] Shouchuan Zhang. The H-Von Neumann regular radical and the Jacobson radical of twisted graded algebra. Acta Math. Hungar., preprint, 1994.
[39] E.C.Dade. Group-graded rings and modules. Math Z. 1980,174: 241-262.
[40] V.G.Kac. Lie Superalgebras. Advances in Mathmatics, 1977,26:8-26.
[41] S.S.Page and M.F.Yousif. Relative injective and chain condition. Comm.Algebra, 1989,17(4): 899-924.
[2] M.Hall.A type of algebraic closure.Ann.of Math, 1939,40: 360-369.
[3] I.Kaplansky.Dual rings.Ann.of Math, 1948,49:689-701.
[4] I.A. Skornjakov.More on quasi-Frobenius rings. Mat.Sb, 1973,92: 518-529.
[5] C.R.Hajarnavis and N.C.Norton.On dual rings and their modules. J.Algebra, 1985,93:253-266.
[6] S.S.Page and Yiqiang Zhou. Quasi-dual rings.Comm.Algebra,2000,28(1): 489-504.
[7] M.E.Sweedler. Hopf Algebras.Benjamin. New York: Benjamin press, 1969.
[8] Shouchuan Zhang. The radicals of Hopf module algebras. Chinese Ann. Mathematics, Ser B, 1997,18(4): 495-502.
[9] 陈维新，张寿传．广义矩阵环的特殊根．浙江大学学报，1995，29(6)：639-646．
[10] 张寿传和陈维新．Г-环的Baer根和根的一般理论．浙江大学学报(自科版)，1991，25：719-724．
[11] W.E.Coppage and J.Luh. Radicals of gamma rings.J.Maath.soc.Japan, 1971,23(1):40-52.
[12] Weixin Chen and shouchuan Zhang. The module theoretic characteriation of special and supernilpotent radicals for Г -rings. Math. Japonica, 1993,38(3): 541-547.
[13] Booth G L. Supernilpotent radical of Г-rings. Acta Math, Hungarica, 1989, 54(3-4):201-208
[14] S.Kyuno.Nobusawa's gammar rings with right and left unities.Math.Japonice, 1980,25: 180-190.
[15] 刘绍学．加法范畴的Baer根和Levitzki根．北京师大学报，1987，4：13-27．
[16] G.L.Booth. On the radicals of Г_N-rings. Math. Japonica, 1987,32(3): 357-372.
[17] Shoaxue Liu.The geometric properties of directed graph and the algebraic poperties of path algebra. Acta mathematica sinica, 1988,31(4): 433-487.
[18] Shouchuan Zhang. The Baer radical of generalized matrix rings. In:Proceedings of the sixth SIAM conference on parallel processing for scientific computing. American, 1993.
[19] 陈维新．Г-环的最大Von Neumann正则理想．浙江大学学报，1981，3:68-72．
[20] Frank W.Anderson and Kent R.Fuller.Rings and Categories of Modules.Springer GTM, 1973.
[21] Nathan Jacobson.Basic aigebra Ⅱ. San Franciso: W.H.Freeman and Company, 1974.
[22] R.Wibauer.Foundations of Module and Ring Theory.Gordon and Breach: Reading, 1991.
[23] Nguyen Viet Dung, Dinh Van Huynh and Patrick F.Smith,etc. Extending Modules.Heinrich Heine University Dsseldorf, 1994.
[24] A.O.AI-attas and N.Vanaja. On ∑ -extending modules.Comm. algebra, 1997,25(8): 2365-2393.
[25] R.Wisbauer. Generalized co-semisimple modules.Comm.Algebra, 1990, 18:4235-4253.
[26] Shenglin Zhu.On rings over which every flat left modules is finitly projective. J.Algebra,1991,139: 311-321.
[27] F.A.Szasz.Radicals of rings.New York:Jahn wiley and Sons, 1982.
[28] Y.Krata and K.Koike. Dual-bimodules and finitely cogenerated modules Math.J.Okayama Univ, 1995, 37:91-98.
[29] W.K.Nicholson and J.K.Park. Principally Quasi-injective Modules. Comm. Algebra, 1999,27(4): 1683-1693.
[30] W.K.Nicholson and M.F.Yoush, Principally injective rings, J.Algebra 174, 77-93(1995).
[31] S.Montgomery. Hopf algebras and their actions on rings, CBMS Number 82, Published by AMS, 1993.
[32] R.J.Blattner, M.Cohen and S.Montgomery. Crossed products and inner actions of Hopf algebras. Transactions of the AMS. 1986,298(2):671-711.
[33] S.Montgomery. Crossed products of Hopf algebras and enveloping algebras, in Perspectives in Ring Theory. Kluwer 1988, 253-268.
[34] Shouchuan Zhang. Braided Hopf algebra. Chang Sha: Hunan Normal University press, 1999.
[35] Miriam Cohen and Davida Fishman. Hopf algebra actions. J.of Algebra, 1986, 100:363-379.
[36] 陈维新和张寿传．有单位元的单Г-环类的根．数学研究，1996，16(2)：263-268．．
[37] A.D.Sands. Radicals and Morita Contexts. J.Algebras, 1973, 24: 335-345.
[38] Shouchuan Zhang. The H-Von Neumann regular radical and the Jacobson radical of twisted graded algebra. Acta Math. Hungar., preprint, 1994.
[39] E.C.Dade. Group-graded rings and modules. Math Z. 1980,174: 241-262.
[40] V.G.Kac. Lie Superalgebras. Advances in Mathmatics, 1977,26:8-26.
[41] S.S.Page and M.F.Yousif. Relative injective and chain condition. Comm.Algebra, 1989,17(4): 899-924.