Definition:Division Ring
Alternative definitions given as separate equally weighted definitions on transcluded pages
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Definition
A division ring is a ring with unity $\left({R, +, \circ}\right)$ such that:
- $\forall x \in R^*: \exists! x^{-1} \in R^*: x^{-1} \circ x = x \circ x^{-1} = 1_R$
where $R^*$ denotes the set of elements of $R$ without the ring zero $0_R$:
- $R^* = R \setminus \left\{ {0_R}\right\}$
That is, every non-zero element of $R$ has a (unique) non-zero product inverse.
Alternative definitions
A division ring is a ring with unity $\left({R, +, \circ}\right)$ such that:
This follows from how a unit is defined.
- $(2): \quad R$ has no proper elements.
This follows from the fact that a unit is not a proper element.
- $(3): \quad R$ has no proper zero divisors.
This follows from the fact that a unit can not be a zero divisor.
Also see
- Results about division rings can be found here.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): $\S 23$
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 6$: Rings and fields
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 55 \ (3)$
