If A , B are finitely generated R-algebras then $A\otimes_RB$ is a finitely generated $R$-algebra.

Question

$A$, $B$ are finitely generated $R$-algebras. $R$ is a commutative ring with $1$. Then how can I show that $A\otimes_RB$ is finitely generated $R$-algebra?

What I have tried: First I have to show that $A\otimes_RB$ is an $R$-algebra.For which I need to show there is a ring structure in $A\otimes_RB$.The natural way of defining the multiplication is $(a\otimes b)(c\otimes d)=(ac\otimes bd)$.Well difineness comes from the universal property.Bt how can I show that $A\otimes_RB$ is finitely generated $R-algebra$.Help me.Thank you.

Answer 1

1. $R[X_1,\dotsc,X_n] \otimes_R R[Y_1,\dotsc,Y_m] = R[X_1,\dotsc,X_n,Y_1,\dotsc,Y_m]$.

2. An $R$-algebra is f.g. iff it admits a surjective homomorphism from some polynomial algebra in finitely many variables.