When is a subgroup of a ring an ideal?

TL;DR

This paper characterizes when subgroups of certain commutative rings are ideals, providing necessary and sufficient conditions, computable criteria, and probability estimates for subgroups being ideals.

Contribution

It offers a comprehensive analysis of subgroup-ideal correspondence in specific rings, including explicit criteria and probabilistic results, advancing understanding in ring theory.

Findings

Necessary and sufficient conditions for subgroups to be ideals in $bZ^d$ and $bZ_{n_i}$ rings.

Computable criteria for subgroups of $bZ imes bZ$ and $bZ_n imes bZ_m$ rings.

Probability estimates that a random subgroup is an ideal.

Abstract

Let $R$ be a commutative ring. When is a subgroup of $(R, +)$ an ideal of $R$? We investigate this problem for the rings $\mathbb{Z}^{d}$ and $\prod_{i=1}^{d} \mathbb{Z}_{n_{i}}$. For various subgroups of these rings we obtain necessary and sufficient conditions under which the above question has an affirmative answer. In the case of $\mathbb{Z} \times \mathbb{Z}$ and $\mathbb{Z}_n \times \mathbb{Z}_m$, 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 $\mathbb{Z}_n \times \mathbb{Z}_m$ is an ideal.