When are minimal faithful modules over algebras unique? Let $A$ be a unital associative algebra over $\mathbb{C}$. We say an $A$-module $V$ is "minimal faithful" if (i) $V$ is faithful and (ii) $V$ does not have a proper submodule that is faithful.
First of all, does a faithful $A$-module exist? If so, is it finite-dimensional / finitely-generated? What conditions must $A$ satisfy such that these are true?
Is a minimal faithful module of $A$ unique up to isomorphism? If not, what are some necessary and sufficient conditions on $A$ such that uniqueness holds? For example, is it enough to assume that $A$ is semisimple and Frobenius?
Bonus: Let $A$ and $V$ be graded by an abelian group. (For example, consider supermodules over superalgebras). How do the answers to the above questions change?
 A: Yes, $A$ has a faithful module. To see this, note that since $1\in A$ satisfies $1a=a$, the left regular module $_AA$ is faithful (and finitely generated by $1\in A$). 
If $A_0={_AA}$ is not minimal, then it contains a proper submodule $A_1$ which is still faithful. Repeating, you can get a descending chain of faithful $A$-modules
$$A_0\supset A_1\supset\cdots$$
If you impose the descending chain condition on $A$ (i.e. make $A$ artinian) then you can find a minimal one. There is no reason why this minimal one needs to be unique, though.
In the case of graded algebras, nothing changes except for the fact that you require everything to be graded...
As for the question about whether a faithful representation needs to be finite dimensional, the answer is no. Take the Heisenberg algebra $H$ which is the $\mathbb{C}$-algebra generated by elements $x,d$ satisfying $dx-xd=1$. A minimal faithful representation of this algebra comes from its action on the polynomial ring $\mathbb{C}[X]$, where $x.p(X)=Xp(X)$ and $d.p(x)=\frac{d}{dx}p(x)$. This representation is infinite dimensional.
