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Concept
Minimal ideal
Summary
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 .
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.
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.
Many standard facts on minimal ideals can be found in standard texts such as , , , and .
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 N2 = {0} or N = eR for some idempotent element e of R .
If N1 and N2 are non-isomorphic minimal right ideals of R, then the product N1N2 equals {0}.
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