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Let $\mathbb{F}$ be a field and let $A$ be an associative unital $\mathbb{F}$-algebra. Is there a criterion to let me know if $A$ is isomorphic to the algebra $\mbox{End}(\mathbf{V})$ of endomorphisms for some $\mathbb{F}$-vector space $\mathbf{V}$? Generalizations to $A$ being an associative unital ring and $\mathbf{V}$ an Abelian group or similar are welcome. Answers for particular cases $\mathbb{F}\in\{\mathbb{R},\mathbb{C}\}$ are also appreciated. Thank you.

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According to this paper:

BRENNER, S. AND BUTLER, M. C. R., Endomorphism rings of vector spaces and torsion-free Abelian groups. J. London Math. Sot. $40 (1965), 183-187.$

Every unital associative algebra, E, can be represented as the algebra $\epsilon( U; \mathbb{K})$ of endomorphisms of a vector space $U$ which leave invariant each member of a set $\mathbb{K}$ of distinct proper subspaces of $U$. (The base field $\phi$ is the same for $U$ as for $E$.) This result was proved by constructing a representation of $E$ with $\dim U = 2 \dim E$ and the cardinal |$\mathbb{K}$| of $\mathbb{K}$ equal to $\gamma + 3$, where $\gamma$ is the cardinal of a set of generators of $E$.

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  • $\begingroup$ This is interesting, but I don't think it really answers the question. You should post it as a comment (although you are not able to post comments on other people's posts yet). Does the paper indicate in any way when you can take $\Bbb K$ empty? On the face of it, the construction you are describing will never be able to do that. $\endgroup$
    – tomasz
    Commented Feb 8, 2017 at 16:50

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