Definition of linear independence in R-module

Question

I am revising module theory over commutative rings with 1 and I have a "soft" question regarding to why don't we define linear independence as follows:

"$v_1,...,v_n$ are linearly independent if $\sum \lambda_i v_i =0 \implies \lambda_i$ non-unit".

This feels more natural to me and is compatible with linear independence when $R$ is a field.

Moreover, it seems that more theorems from linear algebra like "any generating set has a linearly independent subset" can be copied over to the module case. Why isn't this done? Or maybe more precisely, why should we care for things to be linearly independent as per the usual definition when working with modules instead of vector spaces.

Answer 1

It's just a question of terminology and usefulness. As far as terminology, there's the standard definition of linear independence and there's your definition. We need to pick one to be called "linearly independent" and the other needs to be called something else. It just so happens that the standard definition is the one we've picked. Of course this doesn't stop you from inventing a name for your definition and then proving theorems about it.

This brings us to usefulness. Why did the one definition get picked over the other? Well, my experience with representation theory leads me to believe that free modules are a very useful class of modules. One characterization of free modules is that they are modules that have a linearly independent generating set. This is only true for general rings if you use the standard definition of linearly independent.

So that's one reason why the standard definition is useful. By useful, I mean there are theorems that are not explicitly about linear independence but the proofs of those theorems use the concept of linear independence, for example many proofs employ free modules using the characterization above. Are there any examples of why the proposed definition is useful? I don't know of any, which is why I wouldn't consider that concept to be a useful one, presently. This of course could change if you start coming up with proofs that use that concept, but the fact that that hasn't apparently happened yet is why mathematicians "care" more about the standard definition of linear independence.