All modules do have the flat cover by a result from the year 2001. What can be a ring $R$ and a left $R$-Module $M$ which doesn't have (a/the) flat pre-cover ? How can I construct such $R$ and left module $M$ over this $R$ ? Are all flat pre-covers of a fixed $M$ unique up to an iso ?
2026-03-28 00:47:41.1774658861
does the flat pre-cover need to exist for a left module?
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Every flat cover is already a flat precover, because it is just a precover with an extra condition.
I think the extra condition does allow you to prove that the precover is unique. For, if you had two precovers and found arrows going in opposite directions between them, then by composing the two (in both orders) you get an endomorphisms, and the extra condition says these endomorphisms are an isomorphisms. That would mean the original two maps were isomorphisms between the two precovers.
Since Bican, Bashir and Enochs proved that every module has a flat cover, what you are looking for does not exist.