Limit distributions of the length and rank of a module generated by the columns of a random equiprobable matrix over a finite chain ring
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''.
scientific article; zbMATH DE number 1418956