#### 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
##### Abstract

Let $R$ be a commutative ring. When is a subgroup of $\left(R,+\right)$ 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 $ℤ×ℤ$ and ${ℤ}_{n}×{ℤ}_{m}$, 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 ${ℤ}_{n}×{ℤ}_{m}$ is an ideal.

##### Keywords
ring, subgroup, ideal, Mathieu subspace, Goursat
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