Does there exist a division ring without unity?

Question

In abstract algebra I have only ever seen division introduced via multiplicative inverses, namely starting from a ring with unity $R$ and then adding the condition that each element $x$ has an inverse element $x^{-1}$ such that $xx^{-1}=x^{-1}x=1$. But I can also imagine a concept of division without having a unit element, defined as follows:

Let $R$ be a ring with the property that for each ordered pair $(a,b)\in R$ with $b\neq 0$, there exists a unique $c\in R$ such that $a=bc=cb$. Therefore it makes sense to define $a/b:=c$, where $c$ is the unique element corresponding to $(a,b)$ as specified above. Is it possible for such a structure to exist on a ring without unity?

Answer 1

Let $R$ be a ring without unit. Suppose

for each ordered pair $(a,b)\in R$ with $b\neq 0$, there exists a unique $c\in R$ such that $a=bc=cb$.

We claim that $R$ is a ring with unit.

Let $b \in R$, $b \ne 0$. Then there is a unique $e_b$ such that $b = be_b = e_bb$.

We must show that $e_a = e_b$ for all nonzero $a,b$. Then this will be the unit in $R$.

Let $a,b \in R$, both nonzero. There is $c$ so that $a = cb = bc$. So $$ e_b a = e_b b c = b c = a,\qquad ae_b= c b e_b = c b = a. $$ Thus, $e_b$ satisfies the defining property of $e_a$. By the uniqueness, $e_a = e_b$.

Answer 2

On the other hand, it is interesting to have another answer by considering non-associative rings.

A (non-associative) ring is an abelian groups $A$ with a binary operation $*:A\times A\rightarrow A$ with the following properties:

1. $a*(b+c)=(a*b)+(a*c)$.

2. $(a+b)*c=(a*c)+(b*c)$.

A division ring is a ring $A$ such that $A\neq\{0\}$ and for every $a\in A$ with $a\neq0$ the functions $L_a:x\mapsto ax$ and $R_a:x\mapsto xa$ are both bijective.

In this non-associative context, we can have a division ring without unity.

Consider the ring $\overset{*}{\mathbb{C}}$, given by the complex numbers with usual sum and the following multiplication:

$$ x*y=\bar{x}\bar{y}, $$

where $x\mapsto\bar{x}$ denotes the usual conjugation.

Then $\overset{*}{\mathbb{C}}$ is a division ring but does not have a unity element.