I feel like this should be easy, but I cannot quite find a literature reference for this: We know (i.a. from the Kaplansky reference in Does Smith normal form imply PID?) that sufficient for Smith normal form as well as Hermite normal form to work is that the underlying ring be a PID.

I am interested in the case where the ring is $k[t]$, for some field $k$, and all modules involved are $\mathbb N$-graded with the "obvious" grading of $k[t]$. For a matrix $M$ representing a map between two graded $k[t]$-modules $S\to T$, it seems obvious to me that Smith normal form is computable, and about as efficient as one might hope over any ring. The presence of a grading seems to imply one should take some minute care — but the care needed seems to be almost non-existent.

Has anyone dealt with this sort of setting in the literature already? I'd rather have a good reference for this than develop everything in analogy with well-known results myself.

I feel like this should be easy, but I cannot quite find a literature reference for this: We know (i.a. from the Kaplansky reference in Does Smith normal form imply PID?) that sufficient for Smith normal form as well as Hermite normal form to work is that the underlying ring be a PID.

I am interested in the case where the ring is $k[t]$, for some field $k$, and all modules involved are $\mathbb N$-graded with the "obvious" grading of $k[t]$. For a matrix $M$ representing a map between two graded $k[t]$-modules $S\to T$, it seems obvious to me that Smith normal form is computable, and about as efficient as one might hope over any ring. The presence of a grading seems to imply one should take some minute care — but the care needed seems to be almost non-existent.

Has anyone dealt with this sort of setting in the literature already? I'd rather have a good reference for this than develop everything in analogy with well-known results myself.