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Suppose $R$ is a unital semi simple ring, not necessarily commutative. It's known that there are only finitely many isomorphism classes of simple left ideals.

Are these isomorphism classes themselves finite, so that there are only finitely many simple left ideals of any given isomorphism type?

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Are these isomorphism classes themselves finite, so that there are only finitely many simple left ideals of any given isomorphism type?

No, not in general. Let $F$ be a field, and consider the left ideal $L_\lambda$ of matrices of the form $\left[\begin{smallmatrix}\lambda a& a\\ \lambda b & b\end{smallmatrix}\right]$ where $\lambda,a,b \in F$, where $\lambda\neq 0$.

These are all simple left ideals of $M_2(F)$, and they are pairwise distinct for distinct values of $\lambda$. If $F$ is infinite, then there are infinitely many of them.

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