modular ideal

Let $R$ be a ring. A left ideal $I$ of $R$ is said to be modular if there is an $e\in R$ such that $re-r\in I$ for all $r\in R$. In other words, $e$ acts as a right identity element modulo $I$:

$$re\equiv r(modI).$$

A right modular ideal is defined similarly, with $e$ be a left identity modulo $I$.

Remark. If an ideal $I$ is modular both as a left ideal as well as a right ideal in $R$, then $R/I$ is a unital ring. Furthermore, every (left, right, two-sided) ideal in a unital ring is modular, implying that the notion of modular ideals is only interesting in rings without $1$.