In abstract algebra, a **division ring**, also called a **skew field**, is a ring in which division is possible. Specifically, it is a non-trivial ring in which every non-zero 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 is the set of all non-zero 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”.

Read more about Division Ring: Relation To Fields and Linear Algebra, Examples, Ring Theorems, Related Notions

### Other articles related to "ring, division ring, rings, division rings, division":

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**Division Ring**- Related Notions

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**division rings**... While

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... Wedderburn's theorem characterizes simple

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**division ring**... Let D be a

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**Division Ring**

... A

**division ring**is a

**ring**such that every non-zero element is a unit ... A commutative

**division ring**is called a field ... A main example of a

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### Famous quotes containing the words ring and/or division:

“There is no magic decoding *ring* that will help us read our young adolescent’s feelings. Rather, what we need to do is hold out our antennae in the hope that we’ll pick up the right signals.”

—The Lions Clubs International and the Quest Nation. The Surprising Years, III, ch.4 (1985)

“That crazed girl improvising her music,

Her poetry, dancing upon the shore,

Her soul in *division* from itself

Climbing, falling she knew not where,

Hiding amid the cargo of a steamship

Her knee-cap broken.”

—William Butler Yeats (1865–1939)