Finding generalized inverses in the ring of linear transformations of a finite dimensional vector space V over a division ring

Question

Let $L$ be the ring of linear transformations of a finite dimensional vector space $V$ over a division ring $D$. Show that for any $l$ belonging to $L$, then there exists a $u\in L$ such that $lul=l$. Such $u$ is called a generalized inverse.

A question of Jacobson’s basic algebra p173, something related to Morita context or Wedderburn-Artin theorem. It can be related in languages of matrix, and maybe it can be solved by some matrix work, but I want to know the solution following the hint of several former questions.

Answer 1

The following are equivalent:

1. For all $a\in R$, there exists an $x$ such that $axa=a$ (such rings are called von Neumann regular)
2. Every principal right ideal of $R$ is a summand.

Since this ring is isomorphic to a square matrix ring over $D$, it's a semisimple ring: all right ideals are summands. This is perhaps where the Artin-Wedderburn theorem is relevant. Morita theory is a bit too general to mention here: it's an extension of what the Artin-Wedderburn theorem says.

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You can also do it by just brute force. Pick a basis $\beta=\{l(v_1),\ldots, l(v_n)\}$ for $l(V)$, and extend it to a basis of $V$. Define $u$ by declaring that $u(l(v_i))=v_i$ and $l(b)=0$ for the other basis elements.

Then for every $y\in V$, $$ l(y)=\sum \lambda_il(v_i) $$

and now applying $u$

$$ ul(y)=\sum\lambda_iul(v_i)=\sum\lambda_iv_i $$

and applying $l$ again

$$ lul(y)=\sum\lambda_il(v_i)=l(y) $$