symmetric monoidal (∞,1)-category of spectra
Let $R$ be a commutative ring. The category of associative algebras over $R$ is the category
of rings under $R$. If $R$ is a commutative rig, we can do the same with
The tensor product of $R$-algebras has as underlying $R$-module just the tensor product of modules of the underlying modules, $A \otimes_R B$. On homogeneous elements $(a,b) \in A \times B \stackrel{\otimes}{\to} A \otimes_R B$ the algebra structure is given by
We write also $A \otimes_R B$ for the tensor product of algebras.
For commutative $R$-algebras, the tensor product is the coproduct in $Comm Alg_R$:
hence the pushout in Comm Ring? (or Comm Rig?)
For $A$ an associative algebra over a field $k$, write $A$Mod for its category of modules of finite dimension. Then the tensor product of algebras corresponds to the Deligne tensor product of abelian categories $\boxtimes \colon Ab \times Ab \to Ab$:
See at tensor product of abelian categories for more.