Rings $R$ such that every [regular] square matrix with entries in $R$ is equivalent to an upper triangular matrix

Let $$\text{M}_n(R)$$ be the ring of $$n$$-by-$$n$$ matrices with entries in a commutative unital ring $$R$$. Theorem III in

C.R. Yohe, Triangular and Diagonal Forms for Matrices over Commutative Noetherian Rings, J. Algebra 6 (1967), 335-368

provides a characterization of the Noetherian rings $$R$$ with the property that every matrix in $$\text{M}_n(R)$$ is equivalent to a diagonal matrix: It turns out that this is the case if and only if $$R$$ is a direct sum of PIDs and completely primary PIRs, where completely primary'' means a local ring with a nilpotent maximal ideal. With this in mind, here are my questions:

(1) Is there any analogous characterization for the rings $$R$$ with the property that every matrix in $$\text{M}_n(R)$$ is equivalent to an upper triangular matrix? (2) Does the property hold for any choice of $$R$$? (3) And if the answer to the previous question is no, what if we restrict attention to the regular matrices in $$\text{M}_n(R)$$?

[EDIT] The answer to questions (2) and (3) is in the negative, as shown by Mohan in their answer, and that's good to know. On the other hand, I'm really hoping for someone to come up with a reference where it's actually proved that, if $$R$$ is taken from a class of commutative rings that is sufficiently interesting and sufficiently larger than direct sums of PIDs and completely primary PIRs, then every regular square matrix with entries in đť‘… is equivalent to an upper triangular matrix. This would not be a characterization in the vein of Yohe's theorem, but still... [END OF EDIT]

Every matrix $$A \in \text{M}_n(R)$$ can be brought to upper triangular form by elementary row transformations; that is, there exist elementary matrices $$E_1, \ldots, E_k \in \text{M}_n(R)$$ such that $$E_1 \cdots E_k A$$ is an upper triangular matrix. But the elementary matrix corresponding to a row-multiplying transformation need not be invertible in $$\text{M}_n(R)$$; although it is definitely invertible in $$\text{M}_n(\mathcal Q(R))$$ when $$A$$ is regular, with $$\mathcal Q(R)$$ being the total ring of fractions of $$R$$. Unfortunatley, I don't see how this helps answering any of my questions (I'm especially interested in the last one).

Glossary. By a regular matrix'', I mean a regular element in the multiplicative monoid of $$\text{M}_n(R)$$; or equivalently, a matrix $$A \in \text{M}_n(R)$$ whose determinant is a regular element of $$R$$. An element $$a$$ in a (multiplicatively written) monoid $$H$$ is regular (or cancellable) if the functions $$H \to H: x \mapsto ax$$ and $$H \to H: x \mapsto xa$$ are both injective.

• When you say equivalent, do you mean by conjugating with a non-singular matrix? What is your definition of regular matrix? May 22 '20 at 15:41
• @Mohan By "equivalent to", I mean "associated to". Two elements $x$ and $y$ in a monoid $H$ â€” in the present case, the multiplicative monoid of $\text{M}_n(R)$ â€” are associated if $x=uyv$ for some units $u, v \in H$. AFAIK, "equivalent" is still a standard (although unfortunate) term used in linear algebra in place of "associated"; and it's also used by Yohe in the paper cited in the OP. The answer to your other question is along the same lines: By a "regular matrix", I mean a matrix that is a regular element in the multiplicative monoid of $\text{M}_n(R)$; I've added a definition to the OP. May 22 '20 at 16:14
• If you are interested in non-Noetherian rings, the Hermite rings in the sense of Kaplansky, or K-Hermite rings in T. Y. Lam's terminology ("Serre's Problem on Projective Modules") have the triangular reduction property for every $n$. As K-Hermite rings are Bézout rings, Noetherian K-Hermite rings are just principal ideal rings (PIRs), which can be characterized as direct sums of finitely many PIDs and special PIRs (Yohe's complete primary PIRs). Note that K-Hermite = Hermite (f.g. stably free modules are free) + Bézout. May 22 '20 at 20:55
• @SalvoTringali Check the Appendix of Section I.4 in "Serre's Problem on Projective Modules" of T. Y. Lam. You will find even more results on K-Hermite rings there. If you don't have access to it, you can always resort to "Elementary divisors and modules", I. Kaplansky, 1949. See also "Diagonal reduction of matrices over rings" (2012) by Bogdan Zabavsky. May 23 '20 at 8:41
• It is a well-known artefact mostly carried over from the pre-category theory days, so it's no surprise that it appears in a 60s paper. But we know better now! May 23 '20 at 23:57

As Luc Guyot mentioned, check out Kaplansky's paper Elementary Divisors and Modules from 1949.

Kaplansky calls a ring Hermite when every $$1 \times 2$$ matrix is equivalent to a diagonal matrix, and shows that equivalently a ring is Hermite iff for every matrix $$M$$ there exists an invertible matrix $$U$$ such that $$MU$$ is upper triangular.

To get a finite product decomposition result like Yohe's, we obviously need to assume that $$R$$ has finitely many minimal primes.

On the other hand, we have the following

Theorem If $$R$$ is a BĂ©zout ring with finitely many minimal primes then there is a finite set of idempotents $$e_i$$ such that the ring $$e_iR$$ is a Hermite ring with a unique minimal prime and $$R \cong \prod e_iR$$. Hence $$R$$ is Hermite.

This is Theorem 2.2 in Elementary Divisor Rings and Finitely Presented Modules by Larsen, Lewis, and Shores.

Maybe we can say more about the structure of these summands. For example, it is easy to show that in any BĂ©zout ring with a unique minimal prime $$P$$, the ideal $$P$$ is essential unless $$R$$ is a domain. Indeed, if $$I \cap P = 0$$, then $$\operatorname{Ann}(a) = P$$ for any $$a \in I$$ and clearly any $$a \in I$$ is nonzero in every localization of $$R$$. Since $$R$$ locally has totally ordered ideals, this implies that $$P$$ is locally $$0$$, hence $$0$$, i.e. $$R$$ is a domain.

So to sum up

Conclusion Let $$R$$ be a ring with finitely many minimal prime ideals. Then the following are equivalent:

$$\ \ (1)$$ $$R$$ is a finite direct product of BĂ©zout rings each of which is either a domain or has a unique minimal prime ideal which is essential.

$$\ \ (2)$$ Matrices over $$R$$ are equivalent to triangular matrices.

I'm not sure if you can say very much in general about the structure of BĂ©zout rings with essential unique minimal prime ideal. But, here's one observation: A BĂ©zout ring with a unique minimal prime ideal $$P$$ has the property that every non-nilpotent element divides every nilpotent element.

Indeed, let $$b$$ be nilpotent and let $$a$$ be not nilpotent, i.e. $$b \in P$$ and $$a \notin P$$. By BĂ©zoutness, pick $$c,d,u,v,r$$ such that $$ac + bd = r$$, $$ru = a, rv = b$$. Deduce that $$r \notin P, v \in P$$. It follows that $$cv + du -1 \in P$$, and we deduce that $$du$$ is a unit. Therefore $$a \mid b$$.

Note that if $$R$$ is additionally Noetherian, we can easily deduce that $$P$$ is the unique prime ideal of $$R$$, which exactly recovers Yohe's result. To see this, note first that it suffices to assume $$R$$ is local with maximal ideal $$M$$, in which case it has totally ordered ideals and $$\bigcap_n M^n = 0$$, which implies $$M^n \subseteq P$$ for some $$n$$ and thus $$M \subseteq P$$. Without the Noetherian hypothesis, $$R$$ can have infinite Krull dimension.

As for your 3rd question, I don't have much to say off the top of my head except that it seems really hard. From your own observations, any ring which is its own total ring of fractions (i.e. regular elements are units) will have this property, and that is quite a broad class of rings, members of which sometimes have ostensibly very little in common.

• How can a $1\times 2$ matrix (i.e., non-square) matrix be equivalent to a diagonal (thus square) one?
– YCor
May 23 '20 at 23:24
• @YCor A diagonal matrix is just defined as a matrix that is zero off of the main diagonal. That's a reasonable question though. May 23 '20 at 23:27
• There is still one thing that I find very unsatisfactory with K-Hermite rings: I still ignore whether trigonal reduction of every $2 \times 2$ matrices (the topic of this question) implies trigonal reduction of every $1 \times 2$. It might not be the case. Determining the ring with trigonal reduction of $2 \times 2$ matrices would be an even more on-topic answer. (Mohan's answer is a step forward in this direction.) May 24 '20 at 9:29
• @Luc Guyot I agree, what I wrote is really a long comment that answers a question somewhere between (1) and the later edit, which asks for sufficiently general conditions. I'd also be interested to know what trigonal reduction of $2 \times 2$ entails for a ring. I will think on it. Even for domains, intuitively feels like a much weaker property than f.g. ideals. May 24 '20 at 18:38

I will answer only the second and third question in your list, the first is too general and open ended.

Take $$R=K[x,y,z,w]$$, polynomial ring in four variables over a field $$K$$. Take the $$2\times 2$$ matrix $$M$$ consisting of the four variables as entries. It is regular, but it is not equivalent to an upper triangular matrix.

If it is, then by determinant considerations, one of the diagonal entries have to be a non-zero constant since $$\det M$$ is irreducible in $$R$$. We have $$uMv=N$$ with $$u,v$$ invertible, $$N$$ has a non-zero constant entry. But, put all variables equal to zero and then $$M(0)=0$$ and then $$N(0)$$ must be zero, which is impossible, since one entry is a non-zero constant.

• Nice! It's good to see a counterexample to the 2nd and 3rd questions. On the other hand, I'm really hoping for someone to come up with a reference to something like "If $R$ is taken from a class of commutative rings that is sufficiently interesting and sufficiently larger than direct sums of PIDs and completely primary PIRs, then every regular square matrix with entries in $R$ is equivalent to an upper triangular matrix." This would not be a characterization in the vein of Yohe's theorem, but it seems plausible that something along these lines is out there in the literature. May 22 '20 at 17:51
• @SalvoTringali I had suspected that and if you had stressed that, I wouldn't have bothered to answer. For the first, do you have some specific class of rings in mind? May 22 '20 at 18:08
• No, I don't. I'd be happy with any pointer to the literature where the question has been tackled. May 22 '20 at 18:13