Minimal ideal#Generalization

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 {{nowrap|{0} ⊆ K ⊆ N}}, then either {{nowrap|1=K = {0}{{null}}}} or {{nowrap|1=K = N}}.
• N is a simple right R-module.

Minimal ideals are the dual notion to maximal ideals.

Many standard facts on minimal ideals can be found in standard texts such as {{harv|Anderson|Fuller|1992}}, {{harv|Isaacs|2009}}, {{harv|Lam|2001}}, and {{harv|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 $\backslash mathrm\{soc\}(R\_R)$ 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 {{nowrap|1=xR = N}}.
• Brauer's lemma: Any minimal right ideal N in a ring R satisfies {{nowrap|1=N² = {0}{{null}}}} or {{nowrap|1=N = eR}} for some idempotent element e of R {{harv|Lam|2001|loc=p. 162}}.
• If N₁ and N₂ are non-isomorphic minimal right ideals of R, then the product {{nowrap|1=N₁N₂}} equals {0}.
• If N₁ and N₂ are distinct minimal ideals of a ring R, then {{nowrap|1=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 {{harv|Lam|2001|loc=p. 174}}.

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 _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 _RR_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.

{{Reflist}}

• {{citation |last1=Anderson |first1=Frank W. |last2=Fuller |first2=Kent R. |title=Rings and categories of modules |series=Graduate Texts in Mathematics |volume=13 |edition=2 |publisher=Springer-Verlag |place=New York |year=1992 |pages=x+376 |isbn=0-387-97845-3 |mr=1245487 }}
• {{citation |last=Isaacs |first=I. Martin |title=Algebra: a graduate course |series=Graduate Studies in Mathematics |volume=100 |origyear=1994 |publisher=American Mathematical Society |place=Providence, RI |year=2009 |pages=xii+516 |isbn=978-0-8218-4799-2 |mr=2472787}}
• {{citation | last=Lam | first=Tsit-Yuen | title=Lectures on modules and rings | publisher=Springer-Verlag | location=Berlin, New York | series=Graduate Texts in Mathematics No. 189 | isbn=978-0-387-98428-5 | mr=1653294 | year=1999}}
• {{citation |last=Lam |first=T. Y. |title=A first course in noncommutative rings |series=Graduate Texts in Mathematics |volume=131 |edition=2 |publisher=Springer-Verlag |place=New York |year=2001 |pages=xx+385 |isbn=0-387-95183-0 |mr=1838439 }}

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

Category:Abstract algebra

Category:Ring theory

Category:Ideals (ring theory)