TITLE: Let $M$ and $N$ be modules, definition of $M \otimes N$
QUESTION [0 upvotes]: The usual construction is via the quotient by quotienting out the subspace $D$ with these desired properties (WLOG swap x for y) where $x \in X$ and $y \in Y$

$(x + x',y) - (x,y) - (x',y)$
$(ax,y) - a(x,y)$

The notations are little confusing for me. I keep thinking the first property comes out to $(0,-y)$ and that's why we need to quotient it out because the elements of the tensor product should be bilinear maps. Hence, I am sure I can't perform subtraction like that. 
I also read somewhere we are really looking at the basis elements 
$$e_{(x+x',y)}(n,m)  = 1$$ if $(n,m) = (x + x',y)$ and $0$ otherwise. But these still doesn't justify the problem that $1 = e_{(x + x',y)}(x + x',y) = e_{(x + x',y)}(x,y) + e_{(x + x',y)}(x',0) = 0$

REPLY [1 votes]: Your calculations seem to suggest that you believe $f(a,b) + f(c,d) = f(a+c, b+d)$ for a bilinear map $f$, which is incorrect.  The correct property is $f(a,b) + f(c,b) = f(a+c, b)$ (notice that the second component does not change), and a similar equation where the first component does not change.
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