Almost strongly unital rings

Oman, Greg; Senkoff, Evan

doi:10.2140/involve.2023.16.453

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Almost strongly unital rings

Greg Oman and Evan Senkoff

Vol. 16 (2023), No. 3, 453–465

DOI: 10.2140/involve.2023.16.453

Abstract

Recently Oman and Stroud (Involve 13:5 (2020), 823–828) determined all rings $R$ for which every subring of $R$ has an identity (which need not be the identity of $R$ ), calling such rings strongly unital. We extend this work to determine the rings $S$ for which every proper subring of $S$ has an identity, yet $S$ does not, calling such rings almost strongly unital. We conclude by classifying the rings $T$ for which a subring $R$ of $T$ has an identity if and only if there is a subring $S$ of $T$ such that $R ⊊ S ⊊ T$ .

Keywords

Artinian ring, commutative ring, nilpotent element, reduced ring, strongly unital ring

Mathematical Subject Classification

Primary: 13A99

Secondary: 13M99, 13E10

Milestones

Received: 22 January 2022

Revised: 19 July 2022

Accepted: 19 July 2022

Published: 10 August 2023

Communicated by Scott T. Chapman

Authors

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

Evan Senkoff
Department of Mathematics University of Colorado Colorado Springs, CO United States

Vol. 16, No. 3, 2023