Skew field

A skew field, also known as a division ring, is a (not necessarily commutative) ring in which every element has a two-sided inverse. Equivalently, a skew field is a field in which multiplication does not necessarily commute. That is, it is a set $S$ along with two operations, $+$ and $\cdot$ such that:

There are elements $1, 0 \in S$ such that $1 \cdot a = a \cdot 1 = a$ and $a + 0 = 0 + a = a$ for all $a \in S$ . (Existence of additive and multiplicative identities.)
For each $a \in S$ other than 0, there exist elements $a^{-1}, -a \in S$ such that $a\cdot a^{-1} = a^{-1}\cdot a = 1$ and $a + (-a) = (-a) + a = 0$ . (Existence of additive and multiplicative inverses.)
$a + b = b + a$ for all $a, b \in S$ (Commutativity of addition.)
$(a + b) + c = a + (b + c)$ for all $a, b, c \in S$ (Associativity of addition.)
$(a \cdot b )\cdot c = a \cdot (b \cdot c)$ (Associativity of multiplication.)
$a(b + c) = ab + ac$ and $(b + c)a = ba + ca$ (The distributive property.)

Every field is a skew field. The most famous example of a skew field that is not also a field is the collection of quaternions.

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Skew field

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