# 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*. (Isaacs 2009, p. 190)

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:

*N*is nonzero and if*K*is a right ideal of*R*with {0} ⊆*K*⊆*N*, then either*K*= {0} or*K*=*N*.*N*is a simple right*R*-module.

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 (Anderson & Fuller 1992), (Isaacs 2009), (Lam 2001), and (Lam 1999).

- In a ring with unity, maximal right ideals always exist. In contrast, minimal right, left, or two-sided ideals in a ring with unity need not exist.
- The right socle of a ring is an important structure defined in terms of the minimal right ideals of
*R*. - Rings for which every right ideal contains a minimal right ideal are exactly the rings with an essential right socle.
- Any right Artinian ring or right Kasch ring has a minimal right ideal.
- Domains that are not division rings have no minimal right ideals.
- In rings with unity, minimal right ideals are necessarily principal 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*xR*=*N*. **Brauer's lemma:**Any minimal right ideal*N*in a ring*R*satisfies*N*^{2}= {0} or*N*=*eR*for some idempotent element*e*of*R*. (Lam 2001, p. 162)- If
*N*_{1}and*N*_{2}are nonisomorphic minimal right ideals of*R*, then the product*N*_{1}*N*_{2}equals {0}. - If
*N*_{1}and*N*_{2}are distinct minimal ideals of a ring*R*, then*N*_{1}*N*_{2}= {0}. - A simple ring with a minimal right ideal is a semisimple ring.
- In a semiprime ring, there exists a minimal right ideal if and only if there exists a minimal left ideal. (Lam 2001, p. 174)

## 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.

## References

- Anderson, Frank W.; Fuller, Kent R. (1992),
*Rings and categories of modules*, Graduate Texts in Mathematics,**13**(2 ed.), New York: Springer-Verlag, pp. x+376, ISBN 0-387-97845-3, MR 1245487 - Isaacs, I. Martin (2009) [1994],
*Algebra: a graduate course*, Graduate Studies in Mathematics,**100**, Providence, RI: American Mathematical Society, pp. xii+516, ISBN 978-0-8218-4799-2, MR 2472787 - Lam, Tsit-Yuen (1999),
*Lectures on modules and rings*, Graduate Texts in Mathematics No. 189, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98428-5, MR 1653294 - Lam, T. Y. (2001),
*A first course in noncommutative rings*, Graduate Texts in Mathematics,**131**(2 ed.), New York: Springer-Verlag, pp. xx+385, ISBN 0-387-95183-0, MR 1838439