TITLE: What is the definition of 'span' in a module?
QUESTION [7 upvotes]: $\newcommand{\supp}{\operatorname{supp}} \newcommand{\span}{\operatorname{span}}$Let $M$ be a module over a ring $R$ and $S\subset M$
Define $\mathscr{A} = \bigcap\{N\subset M: N \text{ is a submodule of } M , S\subset N\}$
Define $\mathscr{B} = \{\sum_{i\in\text{supp}(f)} f(i)i : f\in R^S , \supp(f) \text{ is finite } \}$.
Here, if $R$ has a unity, then these two sets are the same. However, if $R$ does not contain a unity, then these two sets may not be the same.
Which one should i use for the definition of $\span (S)$?
Moreover, i don't know where a concept of module is useful when $R$ has no unity.. Should I just assume that $R$ has a unity?

REPLY [2 votes]: Quick answers

Yeah, rings without identity and their modules are useful!
The intersection definition is always defined and is simple to think about, so you can use it no matter what. The downside is that it isn't very descriptive about its elements.

Span of a set
If $X$ is a subset of a right $R$ module, then we can indeed come up with intrinsic descriptions of what the submodule generated by $X$ looks like in terms of elements of $X$ and $R$. Adding identity to the ring $R$ can result in a simpler presentation.
You can check that this is a submodule of $M$:
$$
\left\{\sum_{finite}x_j+\sum_{finite} x_ir_i\middle| x_j,x_i\in X, r_i\in R\right\}
$$
(The "finite" of course means that you are only adding finitely many nonzero things.)
Moreover, any submodule containing $X$ would have to contain that submodule, so it is exactly the submodule generated by $X$.
Now, if you require $R$ to have an identity and you include the axiom for modules about $m1=m$ for all $m\in M$, you can see in the description above that the "$\sum x_j$" part is redundant because you can rewrite $x_j$ as $x_j1$ and merge the two summations. (The sum would still be finite, and all the summands are of the form $x_ir_i$ now!) That's why the description simplifies to this for unital modules over rings with identity:
$$
\left\{\sum_{finite} x_ir_i\middle| x_i\in X, r_i\in R\right\}
$$
The summation notation is close to the $\mathscr B$ expression you wrote, but I prefer mine since using functions and supports seems to add unnecessary layers of cognitive load.