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Suppose $k$ is a field and $A$ is a $k$-algebra of dimension no larger than 3. If $A$ is semi-simple, then $A$ can be written as a direct sum of simple $k$-algebras. Further one can find $A$ is commutative by exhausting all the cases.

Without the semi-simple condition, what can we say about $A$? Is it still commutative?

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The upper triangular $2\times 2$ matrix ring over a field is three dimensional and noncommutative.

Here's what I mean, if you're not familiar with it:

$\{\begin{bmatrix}a&b\\0&c\end{bmatrix}\mid a,b,c\in\mathbb{F}\}$

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    @Honghao Ah! Yes then you're exactly correct. The subset of matrices which are zero on the diagonal form a nonzero nilpotent ideal.2012-08-22