Let \(V\) be an \(n\)-dimensional vector space over a field \(K\), and identify \(M_n(K)\) with \(\Endo_K(V)\). Describe the left, right, and two-sided ideals of \(M_n(K)\), and identify the simple left and right ideals geometrically.
Solution
Write multiplication in \(\Endo_K(V)\) as composition.
1. Left ideals are determined by subspaces on which every map vanishes.
For a subspace \(U\le V\), set \[ L_U=\{f\in\Endo_K(V):U\subseteq\ker f\}. \] This is a left ideal because if \(f\) vanishes on \(U\), then so does \(g\circ f\) for every \(g\).
Conversely, every left ideal of \(M_n(K)\) is of this form. In matrix language, left ideals are exactly the sets of matrices whose rows lie in a fixed subspace of \(K^n\); equivalently, after choosing coordinates they are generated by a collection of primitive idempotents.
2. Right ideals are determined by allowed image subspaces.
For a subspace \(W\le V\), set \[ R_W=\{f\in\Endo_K(V):\operatorname{im}f\subseteq W\}. \] This is a right ideal because \(f\circ g\) still has image contained in \(W\). Conversely every right ideal is of this form. In matrix language, its matrices have all columns in a fixed subspace.
3. The only two-sided ideals are \(0\) and \(M_n(K)\).
Let \(I\ne0\) be a two-sided ideal and choose \(0\ne A=(a_{ij})\in I\). For some \(i,j\), \(a_{ij}\ne0\). Multiplying on the left and right by matrix units gives \[ E_{ki}AE_{j\ell}=a_{ij}E_{k\ell}\in I. \] Since \(a_{ij}\) is invertible in \(K\), every matrix unit \(E_{k\ell}\) belongs to \(I\). Hence \(I=M_n(K)\).
4. The simple one-sided ideals correspond to one-dimensional quotients or subspaces.
A nonzero left ideal \(L_U\) is simple exactly when \(U\) is a hyperplane, so \(V/U\) is one-dimensional. Equivalently, a simple left ideal consists of maps factoring through one fixed one-dimensional quotient of \(V\).
A nonzero right ideal \(R_W\) is simple exactly when \(W\) is one-dimensional. Thus simple right ideals consist of maps whose image lies in one fixed line of \(V\).