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. Algebra6(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

anychoice of $R$? (3) And if the answer to the previous question is no, what if we restrict attention to theregularmatrices 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.