Von Neumann regular but not self-injective ring

Question

I want an example of a von Neumann regular ring which is not self-injective.

My thanks go to anybody answering.

Answer 1

Let $R$ be the subalgebra of $K^{\mathbb N}$ (a countable direct product of copies of a field $K$; for simplicity one can choose $K=\mathbb Z/2\mathbb Z$) generated by $1$ and $K^{(\mathbb N)}$ (a countable direct sum of copies of the field $K$), that is, $R$ consists of all sequences $(a_n)_{n\ge 0}$ such that $a_n\in K$ and which are constant from some $n$ on. Then $R$ is a von Neumann regular ring and $K^{\mathbb N}$ is a proper essential extension of $R$, so $R$ is not self-injective.