I understand (i.e. correct me if I'm wrong) that when we talk about formal sums of a set $X$ with coefficients on a unital ring $R$ we refer to expressions of the form
$$
\sum_{x\in X} r_x x
$$
where
- $r_x\in R$ for all $x\in X$,
- $r_x=0$ for almost all $x\in X$,
- the "product" $r_xx$ it's not defined as it is not really a product, but only notation.
And that the set of all such formal sums (i.e. the free $R$-module on $X$) $RX$ is isomorphic to the set of function $X\to R$ with finite support $R^{(X)}$ via the isomorphism
$$
f\mapsto \sum_{x\in X}f(x)x
$$
Now, first thing I can't quite grasp is how the $X$ generates $RX$ process works. For example, if we have the element
$$
r_{x_1}x_1+r_{x_2}x_2+r_{x_3}x_3
$$
then we could say it is obtained by "multiplying" the elements $x_1,x_2,x_3$ with $r_{x_1},r_{x_2},r_{x_3}$ respectively, and then adding $r_{x_1}x_1,r_{x_2}x_2,r_{x_3}x_3$ to each other. However, if the products $r_xx$ are not really defined and are actually pure notational, how do we even get to the first step? (I reckon, nonetheless, that $X$ is embedded in $RX$ in the same way $\mathbb{N}$ is embedded in $\mathbb{Z}$: not really a subset, but isomorphic to a subset).
Also, I get a bit confused with what happens when the ring $R$ is the ring of integers $\mathbb{Z}$ and the set $X$ is actually a group with additive notation. In such case for any element $x\in X$ and any integer $a_x\in\mathbb{Z}$ the expression
$$
a_xx
$$
has the meaning of summing $a_x$-times the element $x$ with itself. So, would that invalidate the only notation part of such "products" in the formal sum? In the case that $X$ is already an abelian group, would that mean that the free abelian group on $X$ is actually $X$ itself? What would happened if the group $X$ had multiplicative notation instead?
Sorry if this is very basic stuff or any of my questions doesn't make much sense but, as you can see, I'm quite confused by all of this. Any textbook you could recommend is much appreciated.
If it is of any help, the context in which all my questions arose is the construction of the tensor product of modules.
THANK YOU 🙂
Best Answer
The notation $rx$ is just one element in the free module (as an element of a set). There is no operation defined to this $rx$ yet. Also, the formal sum $\sum_{x\in X} r_x x$ is only notational, meaning, each of those is just another element as a set element. There is nothing special about them yet either.
As long as we have a set of all such formal sums, we now can define operations: addition and scalar multiplication to make the set a module over $R$.
The answer for your second question is no: $a_xx$ is not summing $a_x$-times $x$ with operation in $X$. I guess the confusing point here is that we have two additions (since you want the operation of $G$ is the addition). But the addition in the module $\mathbb{Z}X$ is not the addition in $X$. To have a better intuition, let take $X$ with mutiplicative operation. Then the free module $\mathbb{Z}X$ will look similar (in term of elements) to a polynomial ring with variables in $X$. (this is called group ring if you are interested).