Submodules $H$ satisfying: "if $ax \in H$ for some non-zero scalar $a$, then $x \in H$."

Question

Suppose $R$ is a commutative ring and that $X$ is an $R$-module.

Question. Is there a term for those $R$-submodules $H$ of $X$ satisfying the following?

• For all $x \in X$, if $ax \in H$ for some non-zero $a \in R$, then $x \in H$.

Examples.

1. If $R$ is a field, then every submodule of $X$ necessarily satisfies the above condition.
2. If $R = \mathbb{Z}$ and $X =\mathbb{Z}$, then the only submodules of $X$ satisfying the condition of interest are $\{0\}_X$ and $X$.
3. If $R = \mathbb{Z}$ and $X =\mathbb{Z}^2$, then the only submodules satisfying the condition of interest apart from $\{0\}_X$ and $X$ are the submodules of the form $(a,b)\mathbb{Z}^2$ for $a,b$ coprime.

The hope is that by intentionally refusing to consider those submodules of $X$ that don't satisfy the above condition, we're more likely to get a poset that behaves like the poset of subspaces of a vector space.

I am also interested in answers of the form: "This is only the correct condition if $R$ is an integral domain... for $R$ a general ring, the correct condition is..." Please back up your claims with one or more illustrative examples.