Difference between revisions of "Division ring"

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==Relation to fields and linear algebra==
 
==Relation to fields and linear algebra==
All fields are division rings; more interesting examples are the non-commutative division rings. The best known example is the ring of [[quaternion]]s '''H'''. If we allow only [[rational number|rational]] instead of [[real number|real]] coefficients in the constructions of the quaternions, we obtain another division ring. In general, if ''R'' is a ring and ''S'' is a [[simple module]] over ''R'', then, by [[Schur's lemma]], the [[endomorphism ring]] of ''S'' is a division ring;<ref>Lam (2001), {{Google books quote|id=f15FyZuZ3-4C|page=33|text=Schur's Lemma|p. 33}}.</ref> every division ring arises in this fashion from some simple module.
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All fields are division rings; more interesting examples are the non-commutative division rings. The best known example is the ring of [[quaternion]]s '''H'''. If we allow only [[rational number|rational]] instead of [[real number|real]] coefficients in the constructions of the quaternions, we obtain another division ring. In general, if ''R'' is a ring and ''S'' is a [[simple module]] over ''R'', then, by [[Schur's lemma]], the [[endomorphism ring]] of ''S'' is a division ring;<ref>Lam (2001), {{Google books|id=f15FyZuZ3-4C|page=33|text=Schur's Lemma|title=Schur's Lemma}}.</ref> every division ring arises in this fashion from some simple module.
  
Much of [[linear algebra]] may be formulated, and remains correct, for (left) [[module (mathematics)|modules]] over division rings instead of [[vector space]]s over fields. Every module over a division ring has a basis; linear maps between finite-dimensional modules over a division ring can be described by [[matrix (mathematics)|matrices]], and the [[Gaussian elimination]] algorithm remains applicable. Differences between linear algebra over fields and skew fields occur whenever the order of the factors in a product matters. For example, the proof that the column rank of a matrix over a field equals its row rank yields for matrices over division rings only that the left column rank equals its right row rank: it does not make sense to speak about the rank of a matrix over a division ring.
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Much of [[linear algebra]] may be formulated, and remains correct, for [[module (mathematics)|modules]] over a division ring ''D'' instead of [[vector space]]s over a field. Doing so it must be specified whether one is considering right or left modules, and some care is needed in properly distinguishing left and right in formulas. Working in coordinates, elements of a finite dimensional right module can be represented by column vectors, which can be multiplied on the right by scalars, and on the left by matrices (representing linear maps); for elements of a finite dimensional left module, row vectors must be used, which can be multiplied on the left by scalars, and on the right by matrices. The dual of a right module is a left module, and vice versa. The transpose of a matrix must be viewed as a matrix over the opposite division ring ''D''<sup>op</sup> in order for the rule {{math|(''AB'')<sup>T</sup> {{=}} ''B''<sup>T</sup>''A''<sup>T</sup>}} to remain valid.
  
The [[center of a ring|center]] of a division ring is commutative and therefore a field.<ref>Simple commutative rings are fields. See Lam (2001), {{Google books quote|id=f15FyZuZ3-4C|page=39|text=simple commutative rings|p. 39}} and {{Google books quote|id=f15FyZuZ3-4C|page=45|text=center of a simple ring|exercise 3.4 on p.45}}.</ref> Every division ring is therefore a [[division algebra]] over its center. Division rings can be roughly classified according to whether or not they are finite-dimensional or infinite-dimensional over their centers. The former are called ''centrally finite'' and the latter ''centrally infinite''. Every field is, of course, one-dimensional over its center. The ring of [[Hamiltonian quaternions]] forms a 4-dimensional algebra over its center, which is isomorphic to the real numbers.
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Every module over a division ring is [[free module|free]]; i.e., has a basis, and all bases of a module have the same number of elements. Linear maps between finite-dimensional modules over a division ring can be described by [[matrix (mathematics)|matrices]]; the fact that linear maps by definition commute with scalar multiplication is most conveniently represented in notation by writing them on the ''opposite'' side of vectors as scalars are. The [[Gaussian elimination]] algorithm remains applicable. The column rank of a matrix is the dimension of the right module generated by the columns, and the row rank is dimension of the left module generated by the rows; the same proof as for the vector space case can be used to show that these ranks are the same, and define the rank of a matrix.
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The [[center of a ring|center]] of a division ring is commutative and therefore a field.<ref>Simple commutative rings are fields. See Lam (2001), {{Google books|id=f15FyZuZ3-4C|page=39|text=simple commutative rings|title=simple commutative rings}} and {{Google books|id=f15FyZuZ3-4C|page=45|text=center of a simple ring|title=exercise 3.4}}.</ref> Every division ring is therefore a [[division algebra]] over its center. Division rings can be roughly classified according to whether or not they are finite-dimensional or infinite-dimensional over their centers. The former are called ''centrally finite'' and the latter ''centrally infinite''. Every field is, of course, one-dimensional over its center. The ring of [[Hamiltonian quaternions]] forms a 4-dimensional algebra over its center, which is isomorphic to the real numbers.
  
 
==Examples==
 
==Examples==
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== References ==
 
== References ==
* {{cite book |last1=Lam |first1=Tsit-Yuen |authorlink1= |last2= |first2= |authorlink2= |title=A first course in noncommutative rings |url= |edition=2 |series=Graduate texts in mathematics |volume=131 |year=2001 |publisher=Springer |location= |isbn=0-387-95183-0 |id= }}
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* {{cite book |last1=Lam |first1=Tsit-Yuen |authorlink1=Tsit Yuen Lam |title=A first course in noncommutative rings |url= |edition=2nd |series=[[Graduate Texts in Mathematics]] |volume=131 |year=2001 |publisher=Springer |location= |isbn=0-387-95183-0 | zbl=0980.16001 }}
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==Further reading==
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* {{cite book | last=Cohn | first=P.M. | authorlink=Paul Cohn | title=Skew fields. Theory of general division rings | zbl=0840.16001 | series=Encyclopedia of Mathematics and Its Applications | volume=57 | location=Cambridge | publisher=[[Cambridge University Press]] | year=1995 | isbn=0-521-43217-0 }}
  
 
==External links==
 
==External links==

Latest revision as of 23:47, 26 November 2014

In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a nonzero ring[1] in which every nonzero element a has a multiplicative inverse, i.e., an element x with a·x = x·a = 1. Stated differently, a ring is a division ring if and only if the group of units equals the set of all nonzero elements.

Division rings differ from fields only in that their multiplication is not required to be commutative. However, by Wedderburn's little theorem all finite division rings are commutative and therefore finite fields. Historically, division rings were sometimes referred to as fields, while fields were called “commutative fields”.

Relation to fields and linear algebra

All fields are division rings; more interesting examples are the non-commutative division rings. The best known example is the ring of quaternions H. If we allow only rational instead of real coefficients in the constructions of the quaternions, we obtain another division ring. In general, if R is a ring and S is a simple module over R, then, by Schur's lemma, the endomorphism ring of S is a division ring;[2] every division ring arises in this fashion from some simple module.

Much of linear algebra may be formulated, and remains correct, for modules over a division ring D instead of vector spaces over a field. Doing so it must be specified whether one is considering right or left modules, and some care is needed in properly distinguishing left and right in formulas. Working in coordinates, elements of a finite dimensional right module can be represented by column vectors, which can be multiplied on the right by scalars, and on the left by matrices (representing linear maps); for elements of a finite dimensional left module, row vectors must be used, which can be multiplied on the left by scalars, and on the right by matrices. The dual of a right module is a left module, and vice versa. The transpose of a matrix must be viewed as a matrix over the opposite division ring Dop in order for the rule (AB)T = BTAT to remain valid.

Every module over a division ring is free; i.e., has a basis, and all bases of a module have the same number of elements. Linear maps between finite-dimensional modules over a division ring can be described by matrices; the fact that linear maps by definition commute with scalar multiplication is most conveniently represented in notation by writing them on the opposite side of vectors as scalars are. The Gaussian elimination algorithm remains applicable. The column rank of a matrix is the dimension of the right module generated by the columns, and the row rank is dimension of the left module generated by the rows; the same proof as for the vector space case can be used to show that these ranks are the same, and define the rank of a matrix.

The center of a division ring is commutative and therefore a field.[3] Every division ring is therefore a division algebra over its center. Division rings can be roughly classified according to whether or not they are finite-dimensional or infinite-dimensional over their centers. The former are called centrally finite and the latter centrally infinite. Every field is, of course, one-dimensional over its center. The ring of Hamiltonian quaternions forms a 4-dimensional algebra over its center, which is isomorphic to the real numbers.

Examples

Ring theorems

Wedderburn's little theorem: All finite division rings are commutative and therefore finite fields. (Ernst Witt gave a simple proof.)

Frobenius theorem: The only finite-dimensional associative division algebras over the reals are the reals themselves, the complex numbers, and the quaternions.

Related notions

Division rings used to be called "fields" in an older usage. In many languages, a word meaning "body" is used for division rings, in some languages designating either commutative or non-commutative division rings, while in others specifically designating commutative division rings (what we now call fields in English). A more complete comparison is found in the article Field (mathematics).

Skew fields have an interesting semantic feature: a modifier (here "skew") widens the scope of the base term (here "field"). Thus a field is a particular type of skew field, and not all skew fields are fields.

While division rings and algebras as discussed here are assumed to have associative multiplication, nonassociative division algebras such as the octonions are also of interest.

A near-field is an algebraic structure similar to a division ring, except that it has only one of the two distributive laws.

Notes

  1. In this article, rings have a 1.
  2. Lam (2001), Template:Google books.
  3. Simple commutative rings are fields. See Lam (2001), Template:Google books and Template:Google books.
  4. Lam (2001), p. 10

See also

References

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Further reading

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External links