nLab modular ideal

Definition

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 $e r - r\in I$).

Examples

• 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.

Properties

• 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.

References

• EoM: modular ideal

• Wikipedia: Modular ideal

category: algebra