Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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MAXIMALITY PRESERVING CONSTRUCTIONS OF MAXIMAL COMMUTATIVE SUBALGEBRAS OF MATRIX ALGEBRA

  • Received : 2010.10.29
  • Published : 2012.03.31
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Abstract

Let (R, $m_R$, k) be a local maximal commutative subalgebra of $M_n$(k) with nilpotent maximal ideal $m_R$. In this paper, we will construct a maximal commutative subalgebra $R^{ST}$ which is isomorphic to R and study some interesting properties related to $R^{ST}$. Moreover, we will introduce a method to construct an algebra in $MC_n$(k) with i($m_R$) = n and dim(R) = n.

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References

  1. W. C. Brown, Two constructions of maximal commutative subalgebras of $n{\times}n$ matrices, Comm. Algebra 22 (1994), no. 10, 4051-4066. https://doi.org/10.1080/00927879408825065
  2. W. C. Brown, Constructing Maximal Commutative Subalgebras of Matrix Rings in Small Dimensions, Comm. Algebra 25 (1997), no. 12, 3923-3946. https://doi.org/10.1080/00927879708826096
  3. W. C. Brown and F. W. Call, Maximal commutative subalgebras of $n{\times}n$ matrices, Comm. Algebra 21 (1993), no. 12, 4439-4460. https://doi.org/10.1080/00927879308824808
  4. R. C. Courter, The dimension of maximal commutative subalgebras of $K_{n}$, Duke Math. J. 32 (1965), 225-232. https://doi.org/10.1215/S0012-7094-65-03219-9
  5. N. Jacobson, Schur's Theorem on Commutative Matrices, Bull. Amer. Math. Soc. 50 (1944), 431-436. https://doi.org/10.1090/S0002-9904-1944-08169-X
  6. Thomas J. Laffey, The Minimal Dimension of Maximal Commutative Subalgebras of Full Matrix Algebras, Linear Algebra and Its Applications 71 (1985), 199-212. https://doi.org/10.1016/0024-3795(85)90247-2
  7. Y. Song, On the Maximal Commutative Subalgebras of 14 by 14 Matrices, Comm. Algebra 25 (1997), no. 12, 3823-3840. https://doi.org/10.1080/00927879708826089
  8. Y. Song, Maximal Commutative Subalgebras of Matrix Algebras, Comm. Algebra 27 (1999), no. 4, 1649-1663. https://doi.org/10.1080/00927879908826519
  9. Y. Song, Notes on the Constructions of Maximal Commutative Subalgebra of $M_{n}$(k), Comm. Algebra 29 (2001), no. 10, 4333-4339. https://doi.org/10.1081/AGB-100106759
  10. D. A. Suprunenko and R. I. Tyshkevich, Commutative Matrices, Academic Press, 1968.