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Representation of an irreducible direct sum of simple modules as tensor product

Let $k$ be a field and $A$ a finite-dimensional algebra over $k$.
Fix an infinite set of simple modules $\{M_i\}_{i\in I}$ for $A$ (one can think of choosing a basis).
There is a well-defined tensor product $\otimes$ on the category of finitely generated left $A$-modules.
Moreover, we have a canonical way to represent each simple module $M$ as an $A$-submodule of $\otimes M$.
Question: Does there exists a basis of $M$ over $k$ such that $M$ is in «canonical form» in the sense that every nonzero simple $A$-module appears as a simple submodule in $\otimes M$ with nonzero multiplicity (equivalently, as a direct summand in the sum of $A$-modules $\oplus M_i$)?

A:

Yes! Write $M$ as a tensor product of a finite number of simple submodules $M_i$, and assume $M_i e 0$. Then it is relatively easy to see that $M_i$ is isomorphic to a direct sum of copies of $M$.
This is the result of a theorem of Peter Gabriel (

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