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Let $R$ be a ring with unit. An element $a$ is called (von Neumann) regular if there exists an element $x\in R$ such that $a=axa$. A ring is called regular if every element of $R$ is regular. I know that finite regular rings are unit regular, that is, for each $a\in R$ there exists a unit $u\in R$ such that $a=aua$.

My question is: Are there examples of finite regular rings that are not semi-simple rings?

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No, because a Noetherian (and of course that includes Artinian and finite) von Neumann regular ring is already semisimple.

One way to see: von Neumann regular means the finitely generated right ideals are summands. Noetherian+VNR means all right ideals are summands.

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