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### Is every polynomial ring over any field regular?

Is every polynomial ring over any field regular?
For a field that is algebraically closed, it is true as any maximal ideal of $k[x_1,...,x_n]$ corresponds to a point $(t_1,...,t_n)$ in ...

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### Does regular field extension preserve regularity?

Let $k$ be an arbitrary field and suppose that $K/k$ is a regular field extension. Let $V$ be regular scheme of finite type over $\text{Spec }k$ (not necessarily smooth). Is it true that $\text{Spec ...

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### A particular Isomorphism of graded algebras over a regular local ring

In Hartshorne's "Algebraic Geometry", the following statement is a weaker form of Theorem 8.21A (e), which he quotes from Matsumuura's book on commutative algebra:
Proposition. Let $R$ be a ...

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### Tensor product of regular ring (with some conditions)

Basically, my question is whether this answer is correct. Here is the point. Let $R$ be a ring, and let $A$ and $B$ be $R$-algebras. Suppose that $A$ is regular and $B \otimes_R B$ is regular too. ...