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Let $A$ be a $K$-algebra and consider the category $\text{mod}A$ of finitely generated left $A$-modules. If $M$ is in $\text{mod}A$, then $\text{add}M$ denotes the full subcategory of $\text{mod}A$ whose objects are direct summands of finite direct sums of copies of $M$. I read that $\text{add}M$ is the smallest additive full subcategory of $\text{mod}A$ containing $M$, but I don't understand why.

It appears to me that the smallest additive full subcategory of $\text{mod}A$ containing $M$ should consist of the trivial module $0$ and finite direct sums of copies of $M$ (and all homomorphisms between these modules). Why do we need to take also their direct summands?

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  • $\begingroup$ Where did you read that? $\endgroup$
    – Wojowu
    Commented Mar 2, 2020 at 21:10
  • $\begingroup$ I read it in the book Elements of the Representation Theory of Associative Algebras, volume 1, in page 184. $\endgroup$
    – user144185
    Commented Mar 2, 2020 at 21:13
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    $\begingroup$ This looks like the smallest idempotent complete additive full subcategory of $\mathrm{mod} A$ containing $M$. $\endgroup$
    – Tobias Fritz
    Commented Mar 2, 2020 at 21:17
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    $\begingroup$ Small remark: in your description of the additive subcat. gen. by $M$, you don't have to mention the trivial module as a special case. It is just an empty direct sum, i.e. when the index set is empty. $\endgroup$
    – Martin Brandenburg
    Commented Mar 2, 2020 at 22:50

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