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For which commutative rings $R$ are there field homomorphisms $L \leftarrow K \to L'$ (not assumed to be algebraic or anything) such that $R \cong L \otimes_K L'$? Is there an intrinsic characterization of these commutative rings? Notice that such commutative rings can be quite nasty; they may have infinite Krull dimension and are often not noetherian. A minimal requirement is that there is a field $L$ which admits a homomorphism $L \to R$. Then we may treat $R$ as an $L$-algebra and ask for which $L$-algebras $L \to R$ there is some isomorphism of $L$-algebras $R \cong L \otimes_K L'$. However, if possible, I would like to see a characterization inside the category of commutative rings. If nothing seems to work, I would like to see at least some necessary conditions.

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