Convention. If $R$ is a relation $X \rightarrow Y$, what I mean is that $R$ is a subsets of $X \times Y$. We say that $X$ is the domain of $R$ and that $Y$ is the codomain. We'll write $x \overset{R}{\mapsto} y$ to mean $(x,y) \in R$. Define also that $R(x) = \{y \in Y : (x,y) \in R\}.$
Let $X$ and $Y$ denote $\mathbb{R}$-modules. Then there is, in general, no meaningful choice of projection map $X \otimes Y \rightarrow X$. There does, however, exist a perfectly good relation $$R : X \otimes Y \rightarrow X$$
defined as follows: it's the smallest relation with the above domain and codomain subject to the constraint that for all finite sets $I$ and all $I$-indexed sequences
- $x : I \rightarrow X$
- $y : I \rightarrow Y,$
we have: $$\sum_{i \in I}x_i\otimes y_i \overset{R}{\mapsto} \sum_{i \in I} x_i.$$
In general $R$ will not be deterministic. In particular, note that $$\sum_{i \in I}a_ix_i\otimes y_i = \sum_{i \in I}x_i\otimes a_iy_i \overset{R}{\mapsto} \sum_{i \in I} x_i.$$
This means that we can scale the $x_i$'s by non-zero elements without impacting the output of $R$.
Question. Is there a good description of the subset of $X$ given by the following expression? $$R\left(\sum_{i \in I}a_ix_i\otimes y_i\right)$$
$R$ is in fact the total relation: every element of $X\otimes Y$ is related to every element of $X$. Indeed, note that for any $x\in X$, $R$ turns the expression $x\otimes 0$ into $x$. So, given any element $t\in X\otimes Y$, we can add a term $x\otimes 0$ to our representation of $t$ to add an arbitrary $x$ to the image of $t$ under $R$, and thus $R$ can map $t$ to every single element of $X$.