von Neumann algebra and matrices Is any matrix algebra regular in the sence of von Neuman? if yes could you tell me about any paper talk about this topic? 
 A: A ring $R$ is Von Neumann regular if $\forall a\in R$ there exists $x\in R$ such that $a=axa$. 
We can show that the ring of all $n\times n$ matrices over a regular ring is regular. You can find the proof in: K.R. Goodearl , Von Neumann regular Rings, pag. 4.
A: If my "matrix algebra" you mean $M_n(F)$ for some field $F$, then yes. Such a ring is semisimple, meaning that every right ideal is a direct summand. You can show that a ring $R$ is von Neumann regular iff every finitely generated right ideal of $R$ is a direct summand of $R$.
In fact, you can even show that $M_n(S)$ is von Neumann regular for any von Neumann regular ring $S$. There are a lot of ways to do this, but the ones that come to mind immediately for me are all categorical. This is because von Neumann regularity is a Morita invariant property, and hence it will be preserved by matrix rings. You could learn about this in textbooks like Lam's Lectures on modules and rings or Anderson and Fuller's Rings and categories of modules, although they are speaking more generally, not just about von Neumann regular rings.
Of course, there are algebras of matrices that aren't von Neumann regular: for example, you have the upper triangular matrices $T_n(F)$ for any $n>2$. It has a nonzero nilpotent ideal, but a von Neumann regular ring will never have such an ideal.
