Could someone provide a good reference to look up the existence and uniqueness of Smith Normal Form (SNF) for a PID?

I have seen it done for Euclidean domains but not for a PIDs. I know the difference is not much, but I would like a good reference. Thanks.

  • 2
    $\begingroup$ The 3 elementary row and column operations are not enough to obtain SNF over a PID. $\endgroup$
    – messi
    May 23, 2013 at 9:40
  • 1
    $\begingroup$ What messi says. If $\gcd(a,b)=1$, then Bezout says that we can find $u,v$ such that $au-bv=1$, and we need to include 2-D transforms of the form $$\pmatrix{a&b\cr v&u\cr}$$ in the list of operations. $\endgroup$ May 23, 2013 at 21:29

2 Answers 2


See Hungerford, Algebra, 1974, Proposition 2.11, page 339.


See Jacobson's Basic Algebra I, chapter 3.

See also the book Finitely Generated Abelian Groups and Similarity of Matrices over a Field by Christopher Norman.


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