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). 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.

Definition

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

• N is non-zero 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 ideals are the dual notion to maximal ideals.

Properties

Many standard facts on minimal ideals can be found in standard texts such as (Anderson Fuller), (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 [math]\displaystyle{ \mathrm{soc}(R_R) }[/math] 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 N² = {0} or N = eR for some idempotent element e of R (Lam 2001).
• If N₁ and N₂ are non-isomorphic minimal right ideals of R, then the product N₁N₂ equals {0}.
• If N₁ and N₂ are distinct minimal ideals of a ring R, then N₁N₂ = {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).

Generalization

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 _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 
• 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 
• 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 
• 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 

External links

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