# Is the tensor product of two commutative semiprime Q-algebras semiprime?

A ring is semiprime if it has no non-zero nilpotents. Let $Q$ denote the rational numbers and $A,B$ be a pair of commutative semiprime $Q$ algebras. Is $A\otimes_Q B$ semiprime? It is well known that the tensor product of two fields in finite characteristic can have nilpotents.

• This is (1) in mathoverflow.net/questions/9436/… , right? (I am still looking for a constructive proof, by the way. The similar-sounding question mathoverflow.net/questions/250040/… has been nicely resolved using topos theory. Maybe you can help constructivizing the non-constructive proof from Milne's AG?) – darij grinberg Dec 14 '16 at 17:55
• What you call semiprime is usually called reduced. The positive answer to your question follows directly from Tag 034N and Tag 030V (3) $\Leftrightarrow$ (5) (since $\mathbb Q$ is perfect). – R. van Dobben de Bruyn Dec 15 '16 at 3:41