I'm a bit confused with an example on tensor products. The lemma intended to illustrate is as following
For M,N $\in$ A-module, with relations $M_0$ and $N_0$, respectively, i.e.
$$M_0 \rightarrow A^{\oplus I} \rightarrow M \rightarrow 0$$ and $$N_0 \rightarrow A^{\oplus J} \rightarrow N \rightarrow 0$$ Then we have $$M_0\otimes A^{\oplus I} + N_0\otimes A^{\oplus J} \rightarrow A^{\oplus I} \oplus A^{\oplus J} \rightarrow M\otimes N \rightarrow 0.$$
The example is $A=\mathcal{C}[x,y]$, $M=N=(x,y)$. I sort of understand that $M_0=N_0=Span_{\mathcal{C}[x,y]}(y e_1 -x e_2)$, $M_0\otimes A^{\oplus 2} + N_0\otimes A^{\oplus 2}=Span(ye_{11}-xe_{12},ye_{12}-xe_{22},ye_{11}-xe_{12},ye_{22}-xe_{22})$ where $e_1$ and $e_2$ are basis for $A^ {\oplus 2}$ and $e_{ij}=e_i\otimes e_j$ are basis for $A^ {\oplus 2} \otimes A^ {\oplus 2}$. Then my confusion is I thought $e_1=(1,0)$ and $e_2=(0,1)$ but then what is $e_{11}$ etc.?
Moreover, it's claimed that the image of $e_{11}$ in $M\otimes N$ is $x^2$. Well, it's sort of intuitive but how to work out this from the lemma? I guess I'm stuck by visualizing $e_{ij}$.
Thanks very much for your help!