Definition:Division Ring/Definition 1

Definition

A division ring is a ring with unity $\struct {R, +, \circ}$ such that:

$\forall x \in R_{\ne 0_R}: \exists! x^{-1} \in R_{\ne 0_R}: x^{-1} \circ x = x \circ x^{-1} = 1_R$

where $R^*$ denotes the set of elements of $R$ without the ring zero $0_R$:

$R_{\ne 0_R} = R \setminus \set {0_R}$

That is, every non-zero element of $R$ has a (unique) non-zero product inverse.

Also see

• Equivalence of Definitions of Division Ring

• Results about division rings can be found here.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $23$. The Field of Rational Numbers
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 55 \ (3)$. The definition of a ring and its elementary consequences
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): division ring
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): division ring (skew field)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): division ring (skew field)