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Let $R$ be a nonzero commutative ring with $1$, such that all finite matrices over $R$ have a Smith normal form. Does it follow that $R$ is a principal ideal domain?

If this fails, suppose we additionally suppose that $R$ is an integral domain?

What can we say if we impose the additional condition that the diagonal entries be unique up to associates?

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    $\begingroup$ Doesn't Smith normal form hold for a principal ideal ring, possibly with zero divisors? I am thinking of $\mathbb{Z}/n\mathbb{Z}.$ $\endgroup$ – Victor Protsak Jul 10 '10 at 6:42
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    $\begingroup$ For a summary of what is known about this problem, see Theorem 2.1 of my paper math.mit.edu/~rstan/papers/snf_survey.pdf. $\endgroup$ – Richard Stanley Feb 25 '19 at 14:09
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The implication is false without the assumption that R is Noetherian, because finite matrices don't detect enough information about infinitely generated ideals.

For example, let R be the ring $$ \bigcup_{n \geq 0} k[[t^{1/n}]] $$ where $k$ is a field (an indiscrete valuation ring). Any finite matrix with coefficients in R comes from a subring $k[[t^{1/N}]]$ for some large $N$, and hence can be reduced to Smith normal form within this smaller PID.

However, the ideal $\cup (t^{1/N})$ is not principal.

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    $\begingroup$ @Tyler: I don't think it affects the rest of your argument, but: to get a valuation ring, don't you want $k[[t^{\frac{1}{n}}]]$ instead of $k[t^{\frac{1}{n}}]$? $\endgroup$ – Pete L. Clark Jul 10 '10 at 18:20
  • $\begingroup$ Yes, you are correct - I added that sentence at the last minute. It is simply a ring with an indiscrete valuation. $\endgroup$ – Tyler Lawson Jul 10 '10 at 19:36
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If every matrix has a Smith normal form, then every finitely generated $R$-submodule $M$ of $R^n$ satisfies $R^n/M$ is a finite direct sum of modules isomorphic to $R/aR$. If $R$ is Noetherian this implies that every finitely generated module is a direct sum of modules of the form $R/aR$. So if $I$ is a maximal ideal of the Noetherian $R$ then $R/I$ is a simple module, so if $R/I\cong R/aR$ then $I=aR$ is principal. So in a Noetherian ring with Smith normal form for all matrices, every maximal ideal is principal. Does this imply that all ideals are principal?....I'm not sure :-)

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    $\begingroup$ For R a domain, it implies that R is one-dimensional regular, hence Dedekind, so every nonzero ideal is a product of maximal ideals, therefore principal itself. $\endgroup$ – user2035 Jul 10 '10 at 7:13
  • $\begingroup$ Thanks a-fortiori: each localization at a maximal ideal is a local ring of height at most one, so $R$ has Krull dimension $\le 1$. $\endgroup$ – Robin Chapman Jul 10 '10 at 7:19
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    $\begingroup$ A commutative noetherian ring whose maximal ideals are principal is indeed a principal ideal ring (even if it is not domain). See Theorem 12.3 of Kaplansky's article "Elementary divisors and modules," Trans. Amer. Math. Soc. 66 (1949), 464-491. $\endgroup$ – Manny Reyes Jul 10 '10 at 14:55
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Work on ring-theoretic generalizations of Hermite/Smith normal forms goes way back, but made it into the mainstream via classic papers by Helmer and Kaplansky. Nowadays such rings are called elementary divisor rings, or rings with elementary divisors (r.e.d.) or Helmer rings, etc. A search on such terms, and for citations of Kap's classic paper [1] should quickly answer all your questions and then some.

[1] I. Kaplansky, "Elementary divisors and modules," Trans. Am. Math. Soc., 66, 464-491. (1949).
http://www.ams.org/journals/tran/1949-066-02/S0002-9947-1949-0031470-3/S0002-9947-1949-0031470-3.pdf

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Such rings are apparently called elementary divisor rings. They are necessarily Bezout rings (i.e. every finitely-generated ideal is principal), but not easy to characterize completely.

The first paper giving a nontrivial sufficient condition (beyond classical case) seems to be

Helmer, Olaf The elementary divisor theorem for certain rings without chain condition. Bull. Amer. Math. Soc. 49, (1943). 225--236, MR

More complete results are in a series of papers starting with

Larsen, Max D.; Lewis, William J.; Shores, Thomas S. Elementary divisor rings and finitely presented modules. Trans. Amer. Math. Soc. 187 (1974), 231--248, MR

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For the sake of completeness, I summarize several important definitions in this subject (all close to the original question) and their main relations, as can be found in the literature (see the references below).

We consider all rings $R$ commutative and unital. All the theory that follows was inspired by the structure theorem for finitely generated modules over PIDs.

Firstly, given any $R$-module $M$, we have a free presentation (exact sequence) $$F_1\rightarrow^g F_0\rightarrow^f M\rightarrow 0$$ with $F_0,F_1$ free modules. If $M$ is finitely presented we can pick $F_0\cong R^n$, $F_1\cong R^m$, and then $g$ can be represented by a matrix $A\in$Mat$_{n,m}(R)$ (for fixed bases of $R^m,R^n$), so that $$M\cong R^n/AR^m.$$ Thus the study of (finite) matrices over $R$ is the same as the study of finitely presented $R$-modules (more in general, the study of infinite matrices is the study of all modules).

When $A$ is equivalent to a (nonsquare) diagonal matrix, $PAQ=$diag$(d_1,\ldots,d_r)$ with $P,Q$ invertible over $R$, we have $$M\cong R^s\oplus R/d_1R\oplus\cdots\oplus R/d_rR,$$ so $M$ is a direct sum of cyclic modules, which are moreover cyclically presented. If in addition $d_i|d_{i+1}$ for all $i$ (i.e., if $A$ has a Smith normal form) then the presentation ideals satisfy $R\supseteq d_1R\supseteq d_2R\supseteq\cdots\supseteq d_rR$.

Now let us abstract the previous ideas with the following definitions. We will call a ring:

  • FGC if all its finitely generated modules are direct sums of cyclic modules.
  • FPC if all its finitely presented modules are direct sums of cyclic modules (this nomenclature is not usual).
  • Elementary divisor (ED) if all its finite matrices have a Smith normal form.
  • Hermite if all its $2\times1$ and $1\times2$ matrices are equivalent to a diagonal matrix. This implies that all matrices have a Hermite normal form (Kaplansky 1949).
  • Bezout if all its finitely generated ideals are principal.
  • A PIR if all its ideals are principal.
  • CF if all finite direct sums of cyclic modules $M$ have a canonical form, i.e., $M\cong R/I_1\oplus\cdots\oplus R/I_n$ with $I_1\supseteq I_2\supseteq\dots\supseteq I_n$.
  • CP if all its finitely presented cyclic modules $C$ have a cyclic presentation, i.e., $C\cong R/cR$ with $c\in R$ (this nomenclature is not usual).

We have the following immediate relations between the definitions:

  • FGC implies FPC by definition.
  • ED implies FPC by the exposition above. Moreover it implies CP and CF-only-for-finitely-presented-modules.
  • ED implies Hermite: All matrices have Smith normal form, in particular the $2\times1$ and $1\times2$ ones.
  • Hermite implies Bezout: The ideal $I$ generated by the elements $a,b\in R$ can be seen as the sum of the components of the image of the matrix $\begin{pmatrix}a&0\\b&0\end{pmatrix}:R^2\rightarrow R^2$, which can be invertibly reduced on the left to the form $\begin{pmatrix}c&0\\0&0\end{pmatrix}$, giving $I=cR$.

We have also less obvious relations between them:

  • Bezout implies CP: Let $C$ be a finitely presented cyclic module, $C\cong mR$. Then $$0\rightarrow\text{Ann}_R(m)\rightarrow R\rightarrow C\rightarrow 0$$ is a short exact sequence with $C$ finitely presented and $R$ finitely generated. By a lemma of Bourbaki, Ann$(m)$ is finitely generated, so a finitely generated ideal, hence Ann$(m)=cR$ since $R$ is Bezout. This comes from (Larsen, Lewis, Shores 1974) (they just refer to Bourbaki's result).
  • FPC implies elementary divisor: This is one of the main results from (Larsen, Lewis, Shores 1974). Note that, as elementary divisor equals FPC+CP+CF-fin.presented, what this is saying is that once all finitely presented modules are direct sums of cyclic modules, said modules can be taken cyclically presented and in canonical form, so that the Smith normal form of the corresponding matrix exists.

So now we have $$\text{FGC}\Rightarrow \text{FPC}=\text{ED}\Rightarrow \text{Hermite}\Rightarrow \text{Bezout}\Rightarrow \text{CP}.$$

In the literature there are also answers to interesting questions:

  • The Noetherian setting: As has also been proved here, an elementary divisor ring is Noetherian if and only if it is a PIR (Uzkov 1963). Since for Noetherian rings all submodules of finitely generated modules are finitely generated (so finitely generated modules are finitely presented), we also have that the Noetherian FGCs are exactly the PIRs. Observe that the existence of zero divisors does not prevent the matrices from having a Smith normal form.
  • Diagonal matrices having SNF: The rings such that every diagonal matrix has a Smith normal form are precisely the Bezout rings (Larsen, Lewis, Shores 1974). So, in Bezout rings, the obstruction for matrices having an SNF is not being equivalent to a diagonal matrix.
  • Sufficient condition for ED: Hermite rings ask for a diagonal form of small matrices, the $2\times1$ and $1\times2$ vectors. We may ask which other sizes are needed to guaraantee that the ring is not only Hermite, but elementary divisor. It turns out that order $2$ is enough: a ring is ED if and only if all its $2\times 2$ matrices are equivalent to a diagonal matrix (Larsen, Lewis, Shores 1974).
  • Conditions for Bezout being Hermite: Not all Bezout rings are Hermite. Bezout domains are Hermite, and it is an open problem to determine if all Bezout domains are ED domains, a question coming from (Helmer 1943). Bezout rings with a finite number of minimal prime ideals are Hermite (Larsen, Lewis, Shores 1974).
  • The classification of FGCs: Due to the work of several people, the (commutative) FGCs have been classified. A ring is FGC if and only if:
    • It is Bezout and fractionally self-injective (Vámos 1979).
    • It is a finite direct sum of maximal valuation rings, almost maximal Bezout rings, and torch rings.
    • It is a finite direct sum of rings $S$ satisfying: $S$ has a unique minimal prime $P$, $S/P$ is an h-local Bezout domain, the ideals contained in $P$ form a chain, and for each maximal ideal $M$ of $S$, $S_M$ is an almost maximal valuation ring.
  • FGC implies CF: As a consequence of the classification theorems it can be shown that all finitely generated modules over an FGC can be presented in canonical form (Wiegand, Wiegand 1977).
  • Classification of CF rings: CF rings have also been classified (Shores, Wiegand 1974). They are the finite direct sums of local rings, h-local domains, and rings $S$ with a unique minimal prime $P$ such that $R/P$ is an h-local domain, $P^2=0$ and every ideal of $S$ contained in $P$ is comparable with every ideal of $S$.

A detailed account on FGC rings can be found in (Brandal 1979).

We can also think about diagonal forms of infinite matrices. In this context it is more usual to speak of stacked bases: given two free $R$-modules $F\leq G$, we say that they have stacked bases if there is a basis $\{x_i\}_{i\in I}$ of $G$ such that $\{r_jx_j\}_{j\in J}$ is a basis of $F$ with $I\subseteq J$ and $r_j\in R$ for all $j\in J$, that is, if a basis of $F$ can be formed by taking multiples of some of the vectors of the basis of $G$. Observe that if $F,G$ have stacked bases then $G/F$ is a direct sum of cyclic modules.

Over a PID, the stacked bases theorem generalizes the structure theorem for not necessarily finitely generated modules: If $R$ is a PID and $F\leq G$ with $G$ free are such that $G/F$ is a direct sum of cyclic modules then $F,G$ have stacked bases (Cohen, Gluck 1970).

To further generalize to integral domains, (Fuchs, Salce 2000) broadens the definition of stacked bases so that the submodule needs not be free, building on the structure theorem for finitely generated modules over Dedekind domains: Given modules $H\leq F$ with $F$ free, they have stacked bases if we can write $F=\bigoplus_{i\in\Lambda_F} J_ix_i$, $H=\bigoplus_{i\in\Lambda_H} I_iJ_ix_i$ with $\Lambda_H\subseteq\Lambda_F$, $J_i$ invertible ideals of $R$ (i.e., the $J_ix_i$ are rank one projective modules), and $I_i$ finitely generated ideals of $R$.

Then Dedekind domains satisfy the stacked bases property for finitely presented modules $F/H$. (Fuchs, Salce 2000) shows that if $R$ is an h-local Prüfer domain and $M$ is a direct sum of cyclic $R$-modules of projective dimension one, then every presentation $0\rightarrow H\rightarrow F\rightarrow M\rightarrow 0$ of $M$ has stacked bases.

References:

  1. The elementary divisor theorem for certain rings without chain condition (1943). Helmer.
  2. Elementary divisors and modules (1949). Kaplansky.
  3. On the decomposition of modules over a commutative ring into a direct sum of cyclic submodules (1963). Uzkov.
  4. Stacked bases for modules over principal ideal domains (1970). Cohen, Gluck.
  5. Elementary divisor rings and finitely presented modules (1974). Larsen, Lewis, Shores.
  6. Rings whose finitely generated modules are direct sum of cyclics (1974). Shores, Wiegand.
  7. Commutative rings whose finitely generated modules are direct sums of cyclics. Wiegand, Wiegand.
  8. Commutative rings whose finitely generated modules decompose (1979). Brandal.
  9. Sheaf-theoretical methods in the solution of Kaplansky's problem (1979). Vámos.
  10. Stacked bases over h-local Prüfer domains (2000). Fuchs, Lee
  11. Rings and things and a fine array of Twentieth Century associative algebra (2004). Faith. Elementary division rings. FGC rings.
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    $\begingroup$ I like your review very much (+1, I didn't know about Uzkov paper and several other references). If you ever aim at reflecting the state of the art for the stacked bases property, you may consider "Finitely Presented Modules Over Semiheriditary Rings" by F. Couchot, 2007 (zero divisors are handled). $\endgroup$ – Luc Guyot Jun 7 '20 at 19:32

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