# How to find Smith normal form of a matrix over regular rings? This post is a wiki. Anyone with karma >750 is welcome to improve it.

Let $A$ be a matrix over ring $^{\mathbb{Z}}/_{6\mathbb{Z}}$. How to find an invertible matrix $P$ and $Q$ such that the matrix $PAQ$ isa Smith Normal Form of $A$?

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Sort by » oldest newest most voted This does not make sense. See Wikipedia

sage: R = Zmod(6)
sage: R in Domains()
False
sage: M=matrix(R,[[4,3],[2,1]])
sage: M.smith_form()
---------------------------------------------------------------------------
Traceback (most recent call last)
TypeError: Smith form only defined over Noetherian integral domains

more

In fact, the error is misleading: it should say that the Smith form makes sense only over Principal ideal rings which are domains (I wrote it like that for emphasis: of course, I mean the Principal Ideal Domains). There is a similar structure theorem for modules over Dedekind domains, but now the torsion-free = projective which is free + fractional ideal. This probably is not implemented. At any rate, this is not Smith normal form (probably, this must be called the Steinitz normal form...).