When is a subgroup of a ring 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.