Let $(R, +, \cdot)$ be a ring and $I \trianglelefteq R$ be an ideal. One of the defining properties of ideals is that: $$\forall x \in I\ \forall r \in R \qquad r \cdot x, x \cdot r \in I$$
Is there a standard name for this property? I have searched for various sources with the definition of ideals but none of the ones I've found give a name, they just write down the property explicitly.
For example, we usually call the property "$\forall x, y \in S \quad x \cdot y \in S$" of a subring $S \le R$ "closure under multiplication" or "closure under the second ring operation". Is there something analogous for the above? I thought of "closure under multiples", but that might be taken to mean "$\forall x \in I \ \forall n \in \mathbb{N} \quad nx \in I$" (where $nx \doteq \underbrace{x + \cdots + x}_{n}$) instead of the above.
Every ideal $I$ of $R$ is also an $R$-submodule of $R$. With respect to this point of view, one could call the property you described "closure under scalar multiplication". Another more expressive way to describe this property could be "stability under scalar multiplication".