#### Vol. 11, No. 5, 2018

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Rings isomorphic to their nontrivial subrings

### Jacob Lojewski and Greg Oman

Vol. 11 (2018), No. 5, 877–883
##### 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 $G\cong ℤ$ or $G\cong ℤ∕〈p〉$ 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 $R$.

##### Keywords
direct sum, integral domain, polynomial ring, quotient field, reduced ring, zero divisor
Primary: 16B99
Secondary: 20K99