Minimal ideal

In the branch of abstract algebra known as ring theory, a minimal right ideal of a ring "R" is a nonzero right ideal which contains no other nonzero right ideal. Likewise, a minimal left ideal is a nonzero left ideal of "R" containing no other nonzero left ideals of "R", and a minimal ideal of "R" is a nonzero ideal containing no other nonzero two-sided ideal of "R". 
In other words, minimal right ideals are minimal elements of the poset of nonzero 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.
Definition.
The definition of a minimal right ideal "N" of a ring "R" is equivalent to the following conditions:
Minimal right ideals are the dual notion to maximal right ideals.
Properties.
Many standard facts on minimal ideals can be found in standard texts such as , , , and .
Generalization.
A nonzero submodule "N" of a right module "M" is called a minimal submodule if it contains no other nonzero submodules of "M". Equivalently, "N" is a nonzero submodule of "M" which is a simple module. This can also be extended to bimodules by calling a nonzero sub-bimodule "N" a minimal sub-bimodule of "M" if "N" contains no other nonzero sub-bimodules.
If the module "M" is taken to be the right "R"-module "R""R", then clearly 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 "R""R". In the case of two-sided ideals, we see that the minimal ideals of "R" are exactly the minimal sub-bimodules of the bimodule "R""R""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.