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Consider two extension fields $K/k, L/k$ of a field $k$. A frequent question is whether the tensor product ring $K\otimes_k L$ is a field. The answer is "no" and this answer is often justified by so …

For those interested in general statements, here is a summary of assumptions under which the canonical morphisms of $A$-modules below are isomorphisms: If $P$ is finitely generated projective: $$P\ …

Ad (1): It is perfectly true that the tensor product of two reduced algebras over a perfect field is reduced. You can find a proof in Bourbaki's Algebra (Chapter V; §15; 4,5), but of course this vener …

Let $k\subset K$ be a completely arbitrary extension of fields. This extension is said to be separable if equivalently a) For all extensions $k\subset L$, the ring $K \otimes _k L$ is reduced. [A r …