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I'm interested in what to call the length of the sequence of ones in the Smith normal form of a matrix. Equivalently (or so it seems to me), this is the rank of the largest submodule of M, on which A acts injectively (by left-multiplication; there should be an equivalent formulation for right-multiplication). For a given matrix A, let us denote this by R(A). Does this have a name?

For a matrix A over a vector space M, this is simply the rank of A: its the largest number of linearly independent rows/columns in A, the dimension of the image of A, the number of non-zero rows in the RREF of A... etcetera. If we consider A over an R-module M for an arbitrary Euclidean domain, things get a bit more complicated. Over a field, the Smith normal form has only 1s on a part of the diagonal, and 0s elsewhere; more generally, however, the coefficients on the diagonal form sequence where each diagonal coefficient evenly divides the one which follows (this being the definition of the Smith normal form).

For the generalization R(A) of 'rank' to the number of 1s in the Smith normal form, we may easily form non-zero matrices A for which R(A) = 0: for instance, the matrix 2I of any dimension over the integers, or over the integers modulo 2N, for N>1.

What is R(A) called?

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I don't know if this helps, but "row reduction" over PIDs is usually called the "Smith Normal Form": ; this includes the case of $\mathbb{Z}/d\mathbb{Z}$ and any Euclidean domain. – Arturo Magidin Feb 1 '11 at 14:33
Thanks for reminding me of this piece of terminology; I will edit the question to use it. – Niel de Beaudrap Feb 1 '11 at 14:40

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