Rings isomorphic to their nontrivial subrings

Lojewski, Jacob; Oman, Greg

doi:10.2140/involve.2018.11.877

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Rings isomorphic to their nontrivial subrings

Jacob Lojewski and Greg Oman

Vol. 11 (2018), No. 5, 877–883

DOI: 10.2140/involve.2018.11.877

Abstract

Let $G$ be a nontrivial group, and assume that $G ≅ H$ for every nontrivial subgroup $H$ of $G$ . It is a simple matter to prove that $G ≅ ℤ$ or $G ≅ ℤ ∕ 〈 p 〉$ 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 $R ≅ S$ for every nontrivial subring $S$ of $R$ .

Keywords

direct sum, integral domain, polynomial ring, quotient field, reduced ring, zero divisor

Mathematical Subject Classification 2010

Primary: 16B99

Secondary: 20K99

Milestones

Received: 15 August 2017

Revised: 11 November 2017

Accepted: 20 November 2017

Published: 2 April 2018

Communicated by Scott T. Chapman

Authors

Jacob Lojewski
Department of Mathematics University of Colorado Colorado Springs, CO United States

Greg Oman
Department of Mathematics University of Colorado Colorado Springs, CO United States

Vol. 11, No. 5, 2018