Let $M,N$ be graded module over a graded commutative ring $S$. Then $f\in S_d,m\in M_k,n\in N_l$, $f\cdot(m\otimes n)=(fm)\otimes n+m\otimes (fn)$.
$\textbf{Q:}$ If $M,N$ are already tensored over $S$, I would expect $f\cdot(m\otimes n)=fm\otimes n=m\otimes fn$ instead. If $f\cdot(m\otimes n)=(fm)\otimes n+m\otimes (fn)$. I would get $f\cdot(m\otimes n)=2(fm)\otimes n$. In particular, take $f=1$, I got $m\otimes n=0$ for any of them. Is this definition of multiplication of $f$ on graded module $M\otimes N$ correct?
Ref. Ueno, Algebraic Geometry 2, Chpt 5, Sec 2.