交换环上的形式矩阵环的零因子和零因子图（英文）
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摘要
本文主要研究交换环R上的形式矩阵环M_n(R;{S_(ijk)})的零因子和零因子图.首先给出了环上形式线性方程组的概念,并且得到了交换环上形式齐次线性方程组有非平凡解的充分必要条件.然后证明了A是M_n(R;{S_(ijk)})的零因子当且仅当A的行列式是R的零因子当且仅当A是R[A]的零因子.最后研究了交换环R上的形式矩阵环M_n(R;{S_(ijk)})的零因子图的性质.
This paper is devoted to zero-divisors of formal matrix ring M_n(R;{s_(ijk)})over a commutative ring R. First, we introduce the notion of left(right) system of formal linear equations over a ring R, and obtain necessary and sufficient conditions for a left(right)homogeneous system of formal linear equations over a commutative ring to have a nontrivial solution. Second, we prove that an element A of M_n(R;{s_(ijk)}) is a zero-divisor if and only if its determinant is a zero-divisor in R, and if and only if A is a zero-divisor in R[A]. Relative concepts and results on system of linear equations and matrix rings over a commutative ring are generalized. Third, we investigate properties of zero-divisor graph of the formal matrix ring M_n(R{s_(ijk)}).
This paper is devoted to zero-divisors of formal matrix ring M_n(R;{s_(ijk)})over a commutative ring R. First, we introduce the notion of left(right) system of formal linear equations over a ring R, and obtain necessary and sufficient conditions for a left(right)homogeneous system of formal linear equations over a commutative ring to have a nontrivial solution. Second, we prove that an element A of M_n(R;{s_(ijk)}) is a zero-divisor if and only if its determinant is a zero-divisor in R, and if and only if A is a zero-divisor in R[A]. Relative concepts and results on system of linear equations and matrix rings over a commutative ring are generalized. Third, we investigate properties of zero-divisor graph of the formal matrix ring M_n(R{s_(ijk)}).
引文
[1] Abyzov, A.N. and Tapkin, D.T., On certain classes of rings of formal matrices, Russ. Math., 2015, 59(3):1-12.
[2] Abyzov, A.N. and Tapkin, D.T., Formal matrix rings and their isomorphisms, Sib. Math. J., 2015, 56(6):9.5.5-967.
[3] Anderson, D.F., Levy, R. and Shapiro, J., Zero-divisor graphs, von Neumann regular rings, and Boolean algebras, J. Pure Appl. Algebra, 2003, 180(3):221-241.
[4] Bozic, I. and Petrovic, Z., Zero-divisor graphs of matrices over commutative rings, Commun. Algebra, 2009,37(4):1186-1192.
[5] Brown, W.C., Matrices over Commutative Rings, New York:Marcel Dekker, 1993.
[6] Cui, C.Q., Properties of formal matrix ring Mn(R; s), Master Thesis, Nanning:Guangxi Teachers Education University, 2014.
[7] Kosan, T., Wang, Z. and Zhou, Y.Q., Nil-clean and strongly nil-clean rings, J. Pure Appl. Algebra, 2016,220(2):633-646.
[8] Krylov, P.A. and Tuganbaev, A.A., Formal matrices and their determinants, J. Math. Sci., 2015, 211(3):341-380.
[9] Redmond, S.P., The zero-divisor graph of a non-commutative ring, Internat. J. Commutative Rings, 2002,1(4):203-211.
[10] Tang, G.H., Cui, C.Q., Zeng, Q.Y. and Zhang, H.B., On zero-divisors of the formal matrix ring M_n(R; s),Journal of Guangxi Teachers Education University:Natural Science Edition, 2014, 31(1):1-6.
[11] Tang, G.H. and Zhou, Y.Q., A class of formal matrix rings, Linear Algebra Appl., 2013, 438(12):4672-4688.
[2] Abyzov, A.N. and Tapkin, D.T., Formal matrix rings and their isomorphisms, Sib. Math. J., 2015, 56(6):9.5.5-967.
[3] Anderson, D.F., Levy, R. and Shapiro, J., Zero-divisor graphs, von Neumann regular rings, and Boolean algebras, J. Pure Appl. Algebra, 2003, 180(3):221-241.
[4] Bozic, I. and Petrovic, Z., Zero-divisor graphs of matrices over commutative rings, Commun. Algebra, 2009,37(4):1186-1192.
[5] Brown, W.C., Matrices over Commutative Rings, New York:Marcel Dekker, 1993.
[6] Cui, C.Q., Properties of formal matrix ring Mn(R; s), Master Thesis, Nanning:Guangxi Teachers Education University, 2014.
[7] Kosan, T., Wang, Z. and Zhou, Y.Q., Nil-clean and strongly nil-clean rings, J. Pure Appl. Algebra, 2016,220(2):633-646.
[8] Krylov, P.A. and Tuganbaev, A.A., Formal matrices and their determinants, J. Math. Sci., 2015, 211(3):341-380.
[9] Redmond, S.P., The zero-divisor graph of a non-commutative ring, Internat. J. Commutative Rings, 2002,1(4):203-211.
[10] Tang, G.H., Cui, C.Q., Zeng, Q.Y. and Zhang, H.B., On zero-divisors of the formal matrix ring M_n(R; s),Journal of Guangxi Teachers Education University:Natural Science Edition, 2014, 31(1):1-6.
[11] Tang, G.H. and Zhou, Y.Q., A class of formal matrix rings, Linear Algebra Appl., 2013, 438(12):4672-4688.