Is there any way to compute the number of minimal left ideals of $M\_n(K)$, the full $n\times n$ matrix ring with entries in the field $K$ ?



Number of minimal left ideals
The lattice of left ideals of $M\_n(K)$ is in bijection with the subspaces of $K^n$, by sending a left ideal $I$ to the subspace of $K^n$ consisting of all rows of elements of $I$ (observe that, from the left, we can do arbitrary row operations). This bijection is order-preserving, and hence the left ideals of $M\_n(K)$ are in bijection with the lines of $K^n$ (or, if you prefer, with the points of $\mathbb P\_{n-1}(K)$).


If $K$ is finite of order $q$, there will therefore be $\frac{q^n - 1}{q - 1}$ minimal left ideals.


PS: This would probably be better suited for Math Stack Exchange.


