Tensor product of a polynomial ring and the quotient of a polynomial ring modulo a prime ideal

Question

Let $S,R$ be two polynomial rings over a field $K$ in a finite number of variables, and let $I$ be a prime ideal of $R$. Then is it true that the tensor product over $K$ between $S$ and $R/I$ is a domain?

Answer 1

Yes. Say $S=K[x_1, \dots, x_n]$, then $S\otimes_K R/I=(R/I)[x_1, \dots, x_n]$. Because $I$ is prime, $R/I$ is a domain, so this is a polynomial ring over a domain, which is again a domain.

What we have used here is that if $A$ is a commutative ring and $B$ is any $A$-algebra, then we have $A[x_1, \dots, x_n] \otimes_A B \cong B[x_1, \dots, x_n]$. This is not that difficult to show. If you're not familiar with this isomorphism, you should try to establish it.