nLab modular ideal


A left (right) ideal II in a (non necessarily unital) ring RR is modular if eR\exists e\in R such that rR\forall r \in R rerIr e - r\in I (respectively errIe r - r\in I).


  • Every left ideal in a unital ring is modular, since we can take e=1e = 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 II in RR the quotient R/IR/I is unital iff II 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.


category: algebra

Last revised on August 23, 2022 at 10:43:23. See the history of this page for a list of all contributions to it.