# large ideal

An ideal $I$ of a ring $R$ is called a large ideal if for every ideal $J$ of $R$ such that $J\neq\{0\}$, $I\cap J\neq\{0\}$

A ring is semiprime iff every large ideal is dense.

Obviously all nontrivial ideal of an integral domain is a large ideal, and the maximal ideal of any non-trivial local ring is a large ideal.

## References

• 1 N.J. Fine, L. Gillman, J. Lambek, ”Rings of Quotients of Rings of Functions”,
Transcribed and edited into PDF from the original 1966 McGill University Press book
(see http://tinyurl.com/24unqshere, Editors: M. Barr, R. Raphael),