Let $R$ commutative ring $M,N$ module
$F(M\times N)$ free $R$ module
$I$ the ideal generated by elements of the form
$$a(m,n)-(m,an), (a \in R, m \in M, n \in N). $$
Is that true any element of $I$ has at least one coefficient not equal to $1$.
What I want to do is below. Let
$L$ abelian module
$f:M\times N \rightarrow L$ an $R$ balanced map.
I want to show $f$ factors through $F(M\times N)/I$.
To show that, I have to show
I can uniquely choose the representative of $F(M\times N)/I$ so that all of its coefficients are $1$ unless it itself is $0$.
Of course I understand how to construct tensor products and the proof that the construction is appropriate. Therefore, what I would like you to tell me here is whether my proof policy is appropriate or not.