What is the definition of 'span' in a module?

Question

$\newcommand{\supp}{\operatorname{supp}} \newcommand{\span}{\operatorname{span}}$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?

Answer 1

About the first question I suppose that's a matter of taste to choose which one of the two submodules above is the span. Anyway there's another possible definition of span: consider the set $$\bar S = S \cup \{ s \in M \mid -s \in S\} \cup \{rs \in M \mid r \in R, s \in S \lor -s \in S\}$$ then we have a module $$\mathcal C= \left\{ \sum_{i=0}^n v_i \mid v_i \in \bar S\right\}$$ this submodule is the submodule of all linear combination can be formed in a module over a (non necessarily unital) ring and so it (at least in my opinion) it deserves the name of span of $S$. It's easy to see that this module is contained in every submodule containing $S$, and so $\mathcal C= \mathcal A$.

About the second question, modules over ring (non necessarily unital) are actions of rings over abelian groups and so are equivalent to representation of rings in abelian groups.

Because there are interesting non unital rings, as one can see in this MO thread, I suppose it's interesting studying action/representation of such rings, and so studying modules over non unital rings can be useful too.

Hope this helps.

Answer 2

Quick answers

1. Yeah, rings without identity and their modules are useful!

2. 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.