Minimal ideal

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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. Template:Harv

Said another way, 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 zero 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 module R is equivalent to the following conditions:

  • If K is a right ideal of R with {0} ⊆ KN, then either K = {0} or K = N.
  • N is a simple right R module.

Minimal right ideals are the dual notion to the idea of maximal right ideals.

Properties

Many standard facts on minimal ideals can be found in standard texts such as Template:Harv, Template:Harv, Template:Harv, and Template:Harv.

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 RR, 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 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 RRR.

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.

References

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External links

http://www.encyclopediaofmath.org/index.php/Minimal_ideal