Rings isomorphic to their nontrivial subrings

Abstract

Let $G$ be a nontrivial group, and assume that $G\cong H$ for every nontrivial subgroup $H$ of $G$. It is a simple matter to prove that $G\cong \mathbb{Z}$ or $G\cong \mathbb{Z}∕\langle p\rangle$ for some prime $p$. In this note, we address the analogous (though harder) question for rings; that is, we find all nontrivial rings $R$ for which $R\cong S$ for every nontrivial subring $S$ of $R$.