Journal of Guangxi Normal University(Natural Science Edition) ›› 2016, Vol. 34 ›› Issue (1): 72-77.doi: 10.16088/j.issn.1001-6600.2016.01.011

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Strong 2-sum Property of Local Rings and Their Extensions

TANG Gaohua, WU Yansheng, ZHANG Hengbin, LI Yu   

  1. School of Mathematics and Statistics,Guangxi Teachers Education University, Nanning Guangxi 530023,China
  • Received:2015-08-23 Published:2018-09-14

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Abstract: An element of a ring R is called to have the strong 2-sum property if it is a sum of two units that commute with each other. And a ring R is called a strong 2-sum ring if every element of R has the strong 2-sum property. Local ring is a very important class of rings. Local rings and their extensions are very important in the research of ring theory,module theory and homological algebra. In this article, a characterization of a local ring to be a strong 2-sum ring is given. Then, the strong 2-sum property of local rings and their power series extension,trivial extension and matrix extension are studied.

Key words: 2-good ring, strong 2-sum ring, local ring, ring extension

CLC Number: 

  • O153.3
[1] WOLFSON K G. An ideal-theoretic characterization of the ring of all linear transformations[J]. American Journal of Mathematics,1953,75(2): 358-386. DOI:10.2307/2372458.
[2] ZELINSKY D. Every linear transformation is a sum of nonsingular ones[J]. Proceedings of the American Mathematical Society,1954,5(4): 627-630. DOI:10.2307/2032048.
[3] VÁMOS P. 2-good rings[J]. The Quarterly Journal of Mathematics,2005,56(3): 417-430. DOI:10.1093/qmath/hah046.
[4] SRIVASTAVA A K. A survey of rings generated by units[J]. Ann Fac Sci Toulouse Math,2010,19(s1): 203-213. DOI:10.5802/afst.1281.
[5] TANG Gaohua,ZHOU Yiqiang. When is every linear transformation a sum of two commuting invertible ones ?[J]. Linear Algebra and Its Applications,2013,439(11): 3615-3619. DOI:1016/j.laa.2013.09.038.
[6] GROVER H K,WANG Zhou,KHURANA D,et al. Sums of units in rings[J]. Journal of Algebra and Its Applications,2014,13(1):1350072. DOI:10.1142/s0219498813500722.
[7] 唐高华,苏华东,易忠.Zn[i]的单位群结构[J]. 广西师范大学学报(自然科学版),2010,28(2): 38-41. DOI:10.16088/j.issn.1001-6600.2010.02.004.
[8] 唐高华,赵淼清,佟文廷. 关于凝聚局部环的正则性[J]. 数学学报,2000,43(4):615-622.
[9] 李先贤,程福长. 非交换 Gr-凝聚半局部环的同调维数[J]. 广西师范大学学报(自然科学版),1998,16(2): 13-17. DOI:10.16088/j.issn.1001-6600.1998.02.003.
[10] TANG Gaohua,ZHOU Yiqiang. Strong cleanness of generalized matrix rings over a local ring[J]. Linear Algebra and Its Applications,2012,437(10): 2546-2559. DOI:10.1016/j.laa.2012.06.035.
[11] YANG Xiande,ZHOU Yiqiang. Strong cleanness of the 2×2 matrix ring over a general local ring[J]. Journal of Algebra,2008,320(6): 2280-2290. DOI:10.1016/j.jalgebra.2008.06.012.
[12] AVNI N,ONN U,PRASAD A,et al. Similarity classes of 3×3 matrices over a local principal ideal ring[J]. Communications in Algebra,2009,37(8): 2601-2615. DOI:10.1080/00927870902747266.
[13] KRYLOV P A. Isomorphism of generalized matrix rings[J]. Algebra and Logic,2008,47(4): 258-262. DOI:10.1007/s10469-008-9016-y.
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