Involve, a Journal of Mathematics Vol. 9, No. 3, 2016

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When is a subgroup of a ring an ideal?

Sunil K. Chebolu and Christina L. Henry

Vol. 9 (2016), No. 3, 503–516

DOI: 10.2140/involve.2016.9.503

Abstract

Let $R$ be a commutative ring. When is a subgroup of $(R, +)$ an ideal of $R$ ? We investigate this problem for the rings $ℤ^{d}$ and $\prod_{i = 1}^{d} ℤ_{n_{i}}$ . In the cases of $ℤ \times ℤ$ and $ℤ_{n} \times ℤ_{m}$ , our results give, for any given subgroup of these rings, a computable criterion for the problem under consideration. We also compute the probability that a randomly chosen subgroup from $ℤ_{n} \times ℤ_{m}$ is an ideal.

Keywords

ring, subgroup, ideal, Mathieu subspace, Goursat

Mathematical Subject Classification 2010

Primary: 13Axx

Secondary: 20Kxx

Milestones

Received: 15 May 2015

Revised: 2 June 2015

Accepted: 17 June 2015

Published: 3 June 2016

Communicated by Kenneth S. Berenhaut

Authors

Sunil K. Chebolu
Department of Mathematics Illinois State University Normal, IL 61790 United States

Christina L. Henry
Department of Mathematics Illinois State University Normal, IL 61790 United States