product of two algebras is still a algebra?

Question

Let $R$ be a commutative ring, $A$ and $B$ two $R$-algebras, is $A\times B$ still an $R$-algebra? If so, what's the relation between $A\times B$ and $A\otimes B$?

Answer 1

The product $A\times B$ is an $R$-algebra, just defining all the operations (including multiplication by elements of $R$) coordinatewise. It doesn't have any particular relation to $A\otimes B$, though.

(If all your rings are commutative, then the two constructions are in a certain sense dual: $A\times B$ is the product of $A$ and $B$ and $A\otimes_R B$ is the coproduct of $A$ and $B$ (in the category of commutative $R$-algebras).)