Editing Division ring (section)

== Relation to fields and linear algebra ==
All fields are division rings, and every non-field division ring is noncommutative. The best known example is the ring of [[quaternion]]s. If one allows only [[rational number|rational]] instead of [[real number|real]] coefficients in the constructions of the quaternions, one obtains another division ring. In general, if {{math|''R''}} is a ring and {{math|''S''}} is a [[simple module]] over {{math|''R''}}, then, by [[Schur's lemma]], the [[endomorphism ring]] of {{math|''S''}} is a division ring;{{sfnp|Lam|2001|loc={{Google books|id=f15FyZuZ3-4C|page=33|text=Schur's Lemma|title=Schur's Lemma}}|ps=}} every division ring arises in this fashion from some simple module.

Much of [[linear algebra]] may be formulated, and remains correct, for [[module (mathematics)|modules]] over a division ring {{math|''D''}} instead of [[vector space]]s over a field. Doing so, one must specify whether one is considering right or left modules, and some care is needed in properly distinguishing left and right in formulas. In particular, every module has a [[basis (linear algebra)|basis]], and [[Gaussian elimination]] can be used. So, everything that can be defined with these tools works on division algebras. [[Matrix (mathematics)|Matrices]] and their products are defined similarly.{{citation needed|date=July 2023}} However, a matrix that is left [[invertible matrix|invertible]] need not to be right invertible, and if it is, its right inverse can differ from its left inverse. (See ''{{slink|Generalized inverse#One-sided inverse}}''.)

[[Determinant]]s are not defined over noncommutative division algebras.  Most things that require this concept cannot be generalized to noncommutative division algebras, although generalizations such as [[quasideterminant]]s allow some results{{what|date=February 2025}} to be recovered.

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 {{math|''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.

Every module over a division ring is [[free module|free]]; that is, it has a basis, and all bases of a module [[Invariant basis number|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 the 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.

Division rings are the only [[ring (mathematics)|ring]]s over which every module is free: a ring {{math|''R''}} is a division ring if and only if every {{math|''R''}}-module is [[Free module|free]].<ref>Grillet, Pierre Antoine. Abstract algebra. Vol. 242. Springer Science & Business Media, 2007</ref>

The [[center of a ring|center]] of a division ring is commutative and therefore a field.<ref>Simple commutative rings are fields. See {{harvp|Lam|2001|loc={{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 one dimensional over its center. The ring of [[Hamiltonian quaternions]] forms a four-dimensional algebra over its center, which is isomorphic to the real numbers.