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It is well-known that matrices over a PID always have Smith normal form as in https://en.m.wikipedia.org/wiki/Smith_normal_form. What about general rings which may not even be noetherian?

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The algorithm requires at least a Bézout domain to express GCDs as linear combinations, and I think it requires some sort of bound on factorizations to prevent chasing some infinite factorizations forever.

The safest thing to do to control factorizations would be to assume it is a UFD, but a Bézout UFD is already a PID.

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