Let $M_{n\times m}(K)$ be the ring of $n\times m$ matrices over the field $K$, $S=M_{n\times n}(K)$, $R=M_{m\times m}(K)$, $P=M_{n\times m}(K)$ $-$ (S,R)-bimodule, $Q=M_{m\times n}(K)$ $-$ $(R,S)$-bimodule.
I believe that $$P\otimes_{R} Q\cong S$$ as $(S,S)$-bimodules. However, I am struggling to show that. I tried to look at $P\otimes_{R} Q$ as a quotient of $P\otimes_K Q$, which is the Kronecker product, but didn't get far.