Structure of self-injective von Neumann regular rings
A right self-injective von Neumann regular ring is uniquely a direct product of rings of types $I$, $II$ and $III$. Additionally, there are two subtypes of $I$ and of $II$ for which the ring can be uniquely expressed as a product.
Reference(s)
K. R. Goodearl. Von Neumann regular rings. (1991) @ Theorem 10.13, p 115
K. R. Goodearl. Von Neumann regular rings. (1991) @ Theorem 10.22, p 120