A left (right) ideal $I$ in a (non necessarily unital) ring $R$ is modular if $\exists e\in R$ such that $\forall r \in R$ $r e - r\in I$ (respectively $e r - r\in I$).
Every left ideal in a unital ring is modular, since we can take $e = 1$.
Every left ideal containing a modular ideal is modular. In particular, any maximal ideal containing given proper modular ideal is modular.
For a two-sided ideal $I$ in $R$ the quotient $R/I$ is unital iff $I$ is modular as both a left and a right ideal.
The Jacobson ideal of a ring coincides with the intersection of all maximal modular ideals.
EoM: modular ideal
Wikipedia: Modular ideal
Last revised on August 23, 2022 at 10:43:23. See the history of this page for a list of all contributions to it.