Let $K$ be a commutative ring and $R$ a $K$-algebra. Consider the $K$-algebra $R^e=R\otimes_K R^{op}$. The category of $R-R$-bimodules is a monoidal category with the tensor product $\otimes_R$ of $R$-bimodules and unit $R$. After identifying the category of $R-R$-bimodules with the category $R^e\text{-Mod}$ of left $R^e$-modules, we get a monoidal product $\boxtimes$ on $R^e\text{-Mod}$ corresponding to $\otimes_R$.
Let $A$ be a $K$-algebra. Let $s:R\rightarrow A$ and $t:R^{op}\rightarrow A$ be $K$-algebra maps. This gives a $K$-algebra map $e:R^e\rightarrow A, r\otimes k \mapsto s(r)t(k)$. This induces a $R^e$-bimodule ${}_e A_e$ structure on $A$.
I have two questions:
- Are the next two left $R^e$-modules isomorphic?
${}_e A_e\otimes_{R^e}(R^e\boxtimes {}_e A)\cong {}_e A_t\otimes_{R^{op}}{}_t A$
- Can you simplify the $R^e$-module ${}_e A_e\otimes_{R^e} R$?