What are modules in $\operatorname{add} T$ explicitly? Let $A$ be a $K$-algebra and $T$ an $A$-module. The category $\operatorname{add} T$ is defined as the smallest additive subcategory of the category $\operatorname{mod} A$ (the category of all finite dimensional $A$-modules) containing $T$. I think that the modules in $\operatorname{add} T$ are direct summands of $\oplus_{i=1}^{d} T$, where $d$ is some integer. For example, the direct summands of $T$ are in $\operatorname{add} T$. Is this true? What are modules in $\operatorname{add} T$ explicitly? Thank you very much.
 A: Looking at the definition of additive category:


*

*$\mathrm{add} T$ contains $T$, and must contain all finitary biproducts, so must also have $T\oplus T$, $ T\oplus T\oplus T$, $T\oplus T\oplus T\oplus T, \dots$.

*We also need $0$ (which is often written as an axiom in its own right, but you can think of it as the biproduct of the empty set of objects, if you like).

*Finally, we need every Hom-set to be an abelian group under addition, which you can check is already true directly (though it's already true a fortiori, because the category we've constructed is a full subcategory of $\mathrm{mod} A$, and $\mathrm{mod} A$ is an abelian category).


(I don't think you mean "direct summands of $T$". This phrase leads me to think of decomposing $T$ itself as $T = M\oplus N$ within the category of $A$-modules - $M$ and $N$ are the summands, and their sum is $T$. I think you mean "direct sums of copies of $T$".)
(Admittedly, I'm not a category theorist, and I'm a little confused about what 'smallest' means here. I have described the category that has fewest objects, in the obvious sense. If you want a category with even fewer morphisms, you can just throw away all morphisms that don't factor through $0$, but that's probably not what's being asked.)
