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. Likewise, a minimal left ideal is a non-zero left ideal of R containing no other non-zero left ideals of R, and a minimal ideal of R is a non-zero ideal containing no other non-zero two-sided ideal of R (Isaacs 2009, p. 190).

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. The reader is cautioned that outside of this context, some posets of ideals may admit the zero ideal, and so the zero ideal could potentially be a minimal element in that poset. This is the case for the poset of prime ideals of a ring, which may include the zero ideal as a minimal prime ideal.

The definition of a minimal right ideal N of a ring R is equivalent to the following conditions:

Minimal ideals are the dual notion to maximal ideals.

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

A non-zero submodule N of a right module M is called a minimal submodule if it contains no other non-zero submodules of M. Equivalently, N is a non-zero submodule of M which is a simple module. This can also be extended to bimodules by calling a non-zero sub-bimodule N a minimal sub-bimodule of M if N contains no other non-zero sub-bimodules.

If the module M is taken to be the right R-module R_R, then the minimal submodules are exactly the minimal right ideals of R. Likewise, the minimal left ideals of R are precisely the minimal submodules of the left module _RR. In the case of two-sided ideals, we see that the minimal ideals of R are exactly the minimal sub-bimodules of the bimodule _RR_R.

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.