In the branch of abstract algebra known as ring theory, a minimal right ideal of a ringR 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 Rordered 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:
N is nonzero and if K is a right ideal of R with, then either or.
Domains that are not division rings have no minimal right ideals.
In rings with unity, minimal right ideals are necessarilyprincipal right ideals, because for any nonzero x in a minimal right ideal N, the set xR is a nonzero right ideal of R inside N, and so.
Brauer's lemma: Any minimal right ideal N in a ring R satisfies or for some idempotent elemente of R.
If N1 and N2 are nonisomorphic minimal right ideals of R, then the product equals.
If N1 and N2 are distinct minimal ideals of a ring R, then
A nonzero submoduleN of a right moduleM 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 rightR-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 moduleRR. 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.
