Minimal ideal

In the branch of abstract algebra known as ring theory, a minimal right ideal of a ring R is a non-zero right ideal which contains no other non-zero right ideal.

In other words, minimal right ideals are minimal elements of the partially ordered set (poset) of non-zero right ideals of R ordered by inclusion.

Many standard facts on minimal ideals can be found in standard texts such as (Anderson & Fuller 1992), (Isaacs 2009), (Lam 2001), and (Lam 1999).

Just as with rings, there is no guarantee that minimal submodules exist in a module.

Minimal submodules can be used to define the socle of a module.