Limit distributions of the length and rank of a module generated by the columns of a random equiprobable matrix over a finite chain ring
type: Item
instance of: scholarly article
title: Limit distributions of the length and rank of a module generated by the columns of a random equiprobable matrix over a finite chain ring
zbMATH Open document ID: 0958.16018
DOI: 10.1007/BF02667046
published in: Cybernetics and Systems Analysis
publication date: 2001-04-02T00:00:00Z
review text: Let \(R\) be a chain ring of cardinality \(q^N\) with radical \(I(R)\), \(\text{GF}(q)\) its residue field and \(A\) an \((n+s)\times n\) matrix whose elements are jointly independent uniformly distributed on \(R\) random variables, \(s\) a fixed integer. If \(M_R\) is the submodule of \(R^{(n+s)}\) generated by the elements of \(A\) there exists a unique vector \(\beta=(\beta_1,\beta_2,\dots,\beta_{N-1})\) of nonnegative integers called the signature of \(M_R\) such that \(M_R=\oplus^{N-1}_{i=0}\beta_i I(R)^i\) [\textit{N. Jacobson}, The theory of rings (1943; Zbl 0060.07302)]. Let \(l_{n,s}\) be the length of \(M_R\), \(s_0=\max\{0,-s\}\). In this article the author derives a formula for the number of submodules \(M_R\) with the signature \(\beta\), uses it to obtain explicit expressions for the probabilities \(P(l_{n,s}=Nn-k)\), \(s_0N\leq k\leq nN\), and \(\lim_{n\to\infty}P(l_{n,s}=Nn- k)\), \(k=s_0N,s_0N+1,\dots\) and obtains also the limit distributions (under \(n\to\infty\)) for the rank of \(M_R\). The third line from the top, p. 726, in the English translation of the paper should be read as follows: ``whose elements are jointly independent and uniformly distributed on the set \(R\) random variables,\dots''.
Mathematics Subject Classification ID: 16L30
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 60C05
zbMATH DE Number: 1418956
zbMATH Keywords: chain rings
zbMATH Keywords: random variables
zbMATH Keywords: lengths
zbMATH Keywords: numbers of submodules
zbMATH Keywords: probabilities
zbMATH Keywords: limit distributions
author: A. N. Alekseychuk
reviewed by: Feliks R. Bobovich
MaRDI profile type: Publication
cites work: Limit distribution of the number of solutions of a system of stochastic linear homogeneous equations with a uniform coefficient matrix over a finite local ring
cites work: Q4402086
cites work: A Canonical Set For Matrices Over a Principal Ideal Ring Modulo m
cites work: Enumeration of canonical matrices under row equivalence over a principal ideal domain modulo \(p^ n\)
cites work: Q3765777
cites work: Q3964640
Wikidata QID: Q126226927
full work available at URL: https://doi.org/10.1007/bf02667046
OpenAlex ID: W1991123996
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description: scientific article; zbMATH DE number 1418956