Let $\mathcal{A}$ be an essentially small abelian category. We prove that if$\mathcal{A}$ admits a generator $M$ with ${\rm End}_{\mathcal{A}}(M)$ rightartinian, then $\mathcal{A}$ admits a projective generator. If $\mathcal{A}$ isfurther assumed to be Grothendieck, then this implies that $\mathcal{A}$ isequivalent to a module category. When $\mathcal{A}$ is Hom-finite over a field$k$, the existence of a generator is the same as the existence of a projectivegenerator, and in case there is such a generator, $\mathcal{A}$ has to beequivalent to the category of finite dimensional right modules over a finitedimensional $k$-algebra. We also show that when $\mathcal{A}$ is a lengthcategory, then there is a one-to-one correspondence between exact abelianextension closed subcategories of $\mathcal{A}$ and collections ofHom-orthogonal Schur objects in $\mathcal{A}$.
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