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.

13 pages, to appear in Involve