Is there such a thing as a "noncommutative field"?

Question

A field $F$ is a set that is essentially a commutative group in two ways:

1. $(F,+)$ is a commutative group with an identity element we call $0$.
2. $(F\setminus\{0\},\times)$ is a commutative group with an identity element we call $1$.
3. These two operations work together under some kind of distributive property: $a\times (b + c) = (a\times b) + (a\times c)$

This got me thinking, what if a set $G$ had these same three properties but we dropped the conditions of commutativity for one (or both) of the binary operations. Is this something that is interesting to algebraists? Does such a set not even exist? Do these kinds of object already some name that I am unaware of?

Any insight on this matter would be welcomed.

Answer 1

(EDIT: this only addresses the noncommutative multiplication part of the question.)

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Sure, these are called division rings (or skew fields). There is a ton known about them, but here are a couple fun facts:

• Every finite division ring is in fact a field; this is Wedderburn's little theorem.

• Every semisimple ring is "built out of" division rings, in the sense that it is a product of finitely many rings of the form $M_n(D)$ with $n\in\mathbb{N}$ and $D$ a division ring; this is the Artin-Wedderburn theorem.

• If $D$ is a division ring, then the center of $D$ is a field $d$, and $D$ is naturally an algebra over $d$. In case $D$ is finite dimensional over its center, we get that $D$ is a central simple algebra over $d$. Conversely, any finite-dimensional central simple algebra $A$ over a field $k$ is "very similar to" (= Morita-equivalent to) a unique-up-to-isomorphism division algebra $D$ with center $k$ and $dim_k(D)<\infty$. This is part of the (classical) theory of Brauer groups.

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Meanwhile, I don't know much about noncommutative addition, but see the notion of near-rings. (Note, though, that distributivity gets messier if addition isn't commutative.)

Answer 2

I believe that you are looking for the concept of a near-ring, i.e. a set together with two binary operations, none of which is necessarily commutative, but still satisfy a distributive property. See e.g. https://en.wikipedia.org/wiki/Near-ring

Near-rings arise by considering a (not necessarily abelian) group $G$ and the set $N$ of all functions $f\colon G\to G$. Then addition can be defined on $N$ in the usual way, which will be commutative iff $G$ is. Multiplication is also defined in the usual way, i.e. as a composition of functions.