Representations of superalgebras If I have a superalgebra $A$ and consider the category of finite-dimensional $A$-modules.
Is this category the same as the category of finite-dimensional $A$-modules which are super-vector spaces? 
I.e. is every representation of $A$ a super-vector space?

It's a matter of definition. A super algebra is nothing but a particular case of a graded algebra; when we speak of modules over a graded ring we typically mean graded modules. This grading is given by the same group that gives the algebra it's grading (in this case $\mathbb{Z}/2\mathbb{Z}$), and the multiplication is required to respect that grading. 
So the answer is yes, you are correct by definition. 
