@HeinrichD: But in YCor's example there is no injection $\mathbb{Z} \to M_i$, right? It therefore is only a counterexample for the claim "If two modules over a reduced ring have annihilators whose elements are all nilpotent, then their tensor product has this property too", not to the more specific claim involving given maps.

@HeinrichD: I've been following this thread and your other one with much interest. I believe that the techniques explained in my notes indeed suffice to settle the other question; I'll try to write up an answer tomorrow. The key is to use that the structure sheaf of the spectrum of a reduced ring looks like a field from the internal point of view. Therefore finitely generated modules are not not free.

@HeinrichD: Thanks! I'd be willing to accept this as an answer. You can show that the decomposition is unique by verifying that irreducible elements are prime; this is possible with Euclid's lemma (which holds in GCD domains and which admits a constructive proof). Furthermore, one can show that a domain is an UFD in their sense if and only if it is a UFD in the sense of my question and associatedness of irreducible elements is decidable.

A direct reference is cse.chalmers.se/~coquand/sitesur.pdf. Also the slides located at cse.chalmers.se/~coquand/FISCHBACHAU are very good.

Some more examples of the technique explained by Simon are in Section 11.4 of these rough notes of mine.

Dear Fredrik, let me add that I think that your work on Arb is extremely valuable! A free (as in freedom) and curated software library for interval arithmetic is very useful. Also I've been enjoying your insightful blog posts.

If you instruct the person writing down the polynomial to only use nonnegative coefficients, then in fact two evaluations (at integral points) suffice for you to uniquely infer the polynomial, irrespective of its degree.

I fail to see how this is true. I don't see that $A$ and $B$ can manage to find the numbers in case that the numbers are $3$ and $4$. This is even assuming that it's common knowledge that the given numbers are distinct. Without this simplification, I believe that there's a smaller counterexample: the numbers $3$ and $3$. What am I missing?

I don't have time to look at the question right now, but will do so later this day or tomorrow. Meanwhile, in addition to the very useful resources cited by Andrej, you can also have a look at the answers (and the references listed therein) of mathoverflow.net/questions/222923/….

You might like the paper Descent in triangulated categories by Paul Balmer (even if you aren't interested in triangulated categories). It has a very readable introduction in which the basic setup of monadic descent theory is recalled.

An excellent introduction and reference to the $K$-group of coherent sheaves is Manin's 1969 classic Lectures on the $K$-functor in algebraic geometry.