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On a matrix nilpotent filter. (English. Russian original) Zbl 1260.13008
Math. Notes 92, No. 2, 176-185 (2012); translation from Mat. Zametki 92, No. 2, 192-201 (2012).
Consider the \(n\times n\) matrix \(M=(x_{ij})_{i,j=1}^n\) whose entries are \(n^2\) commuting indeterminants \(X=\{x_{ij} \mid 1\leq i,j\leq n\}\). The entries \(f_{ij}^{(s)}\) of the power \(M^s\) are used to define homogeneous ideals \(F^{(s)}= ( f_{ij}^{(s)} \mid 1\leq i,j\leq n)\) in the polynomial ring \(k[X]\) for a field \(k\). This definition yields a decreasing chain of ideals, the so-called matrix nilpotent filter on \(k[X]\). The author studies the combinatorial characteristics of these ideals, with a particular focus on \(n=2\).
MSC:
13A15 Ideals and multiplicative ideal theory in commutative rings
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[1] V. A. Ufnarovskii, ”Combinatorial and asymptotic methods in algebra,” in Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Current Problems in Mathematics. Fundamental Directions, Vol. 57: Algebra-6 (VINITI, Moscow, 1990), pp. 5–177 [in Russian].
[2] D. Cox, J. Little, and D. O’shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 2nd ed. (Springer-Verlag, New York, 1997; Mir,Moscow, 2000).
[3] L. M. Shifner, ”On the power of a matrix,” Mat. Sb. 42(3), 385–394 (1935).
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