I am new to the world of "grading", and my question is about the definition of the graded vector space. For definitions of these two, see graded vector space and graded module (they are basically what I've learned from).
We know that a vector space is a $\mathbb{k}$-module, where $\mathbb{k}$ is a field. So, is a graded $\mathbb{k}$-vector space just the same as a graded $\mathbb{k}$-module? I noticed that in the example of graded module, it mentioned that a graded vector space is a graded module over a field (with the field having trivial grading). But does the definition always requires us using the trivial grading of $\mathbb{k}$? What if $\mathbb{k}$ has a nontrivial grading, for example, $\mathbb{C}=\mathbb{R}\oplus i\mathbb{R}$? Can we use the nontrivial one (maybe to make it interesting)?
Further explanation for my thoughts: As I found that the definition of a graded vector space includes just only the decomposition but almost nothing else, it's a bit weird: what is the meaning of such decomposition if there is no further interaction between the components? For example, the polynomial ring has a natural grading derived from the multiplication of polynomials, so it is a graded ring and interesting. Even the graded module emphisizes the grading structure of the underlying ring. But I cannot see such an "interation" structure in the original definition of graded vector spaces, it somehow seems "trivial" and I feel that there is something I've missed. Does this structure have something further to do with the tensor product or linear transform, something like that to make it not that "trivial"?
Concrete examples are greatly welcome! p.s. I am especially interested in $\mathbb{Z}_2$-graded objects as they come from physics. $\mathbb{Z}_2$-graded rings have a explicit interesting structure while $\mathbb{Z}_2$-graded vector spaces seem not that clear to me at this moment. What is the relation between them? It would be perfect if one could touch upon this specific example. Thanks in advance!