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Precisely, if an R-module M has a finite presentation, and Rk → M is some unrelated surjection (k finite), is the kernel necessarily also finitely generated? Basically I want to believe I can ...

I'm wondering about the question "If we have a finitely presented __, is it necessarily finitely presented with respect to any finite generating set for it?" I know this is true for groups and ...

Let $\mathcal C$ be a locally finitely-presentable category, and let $X$ be a finitely-presentable object of $\mathcal C$. Question: Can there exist a nontrivial idempotent on $X$ whose fixed points ...

EDIT: The original question had a trivial answer: it's just a coproduct. New question below New Question: As shown below, in the category of commutative unital rings, the coproduct of a ring $R$ with $...