In a Commutative Ring, is Addition Necessarily Commutative?

In order to generalize rings to structures with noncommutative addiiton, one cannot simply delete the axiom that addition is commutative, since, in fact, other (standard) ring axioms force addition to be commutative (Hankel, 1867 [1]). The proof is simple: apply both the left and right distributive law in different order to the term $\rm\:(1\!+\!1)(x\!+\!y),\:$ viz.

$$\rm (1\!+\!1)(x\!+\!y) = \bigg\lbrace \begin{eqnarray}\rm (1\!+\!1)x\!+\!(1\!+\!1)y\, =\, x \,+\, \color{#C00}{x\!+\!y} \,+\, y\\ \rm 1(x\!+\!y)\!+1(x\!+\!y)\, =\, x\, +\, \color{#0A0}{y\!+\!x}\, +\, y\end{eqnarray}\bigg\rbrace\:\Rightarrow\: \color{#C00}{x\!+\!y}\,=\,\color{#0A0}{y\!+\!x}\ \ by\ \ cancel\ \ x,y$$

Thus commutativity of addition, $\rm\:x+y = y+x,\:$ is implied by these axioms:

$(1)\ \ *\,$ distributes over $\rm\,+\!:\ \ x(y+z)\, =\, xy+xz,\ \ (y+z)x\, =\, yx+zx$

$(2)\ \, +\,$ is cancellative: $\rm\ \ x+y\, =\, x+z\:\Rightarrow\: y=z,\ \ y+x\, =\, z+x\:\Rightarrow\: y=z$

$(3)\ \, +\,$ is associative: $\rm\ \ (x+y)+z\, =\, x+(y+z)$

$(4)\ \ *\,$ has a neutral element $\rm\,1\!:\ \ 1x = x$

Said more structurally, recall that a SemiRing is that generalization of a Ring whose additive structure is relaxed from a commutative Group to merely a SemiGroup, i.e. here the only hypothesis on addition is that it be associative (so in SemiRings, unlike Rings, addition need not be commutative, nor need every element $\rm\,x\,$ have an additive inverse $\rm\,-x).\,$ Now the above result may be stated as follows: a semiring with $\,1\,$ and cancellative addition has commutative addition. Such semirings are simply subsemirings of rings (as is $\rm\:\Bbb N \subset \Bbb Z)\,$ because any commutative cancellative semigroup embeds canonically into a commutative group, its group of differences (in precisely the same way $\rm\,\Bbb Z\,$ is constructed from $\rm\,\Bbb N,\,$ i.e. the additive version of the fraction field construction).

Examples of SemiRings include: $\rm\,\Bbb N;\,$ initial segments of cardinals; distributive lattices (e.g. subsets of a powerset with operations $\cup$ and $\cap$; $\rm\,\Bbb R\,$ with + being min or max, and $*$ being addition; semigroup semirings (e.g. formal power series); formal languages with union, concat; etc. For a nice survey of SemiRings and SemiFields see [2]. See also Near-Rings.

[1] Gerhard Betsch. On the beginnings and development of near-ring theory. pp. 1-11 in:
Near-rings and near-fields. Proceedings of the conference held in Fredericton, New Brunswick, July 18-24, 1993. Edited by Yuen Fong, Howard E. Bell, Wen-Fong Ke, Gordon Mason and Gunter Pilz. Mathematics and its Applications, 336. Kluwer Academic Publishers Group, Dordrecht, 1995. x+278 pp. ISBN: 0-7923-3635-6 Zbl review

[2] Hebisch, Udo; Weinert, Hanns Joachim. Semirings and semifields. $\ $ pp. 425-462 in: Handbook of algebra. Vol. 1. Edited by M. Hazewinkel. North-Holland Publishing Co., Amsterdam, 1996. xx+915 pp. ISBN: 0-444-82212-7 Zbl review, AMS review

The three basic properties of a ring are;

• The set under addition makes an abelian group,
• Multiplication is associative, and
• Left and right distributive laws hold.

Thus, by definition of "abelian group", the addition must be commutative. Hopes that help.

The definition of a ring is that it has two binary operations, $+$ and $\cdot$. The $+$ operation forms an abelian group and $\cdot$ need only be associative. The distributive laws need hold. However, notice that the distributive laws force $+$ to be abelian when $R$ has $1$!

$$(1+1)(x+y)=1(x+y)+1(x+y)=x+y+x+y$$

and

$$ (1+1)(x+y)=(1+1)x+(1+1)y=x+x+y+y $$ so that $x+y+x+y=x+x+y+y$ then adding $-x$ and $-y$ ($+$ forms a group so has inverses $-x,-y$) on the left and right yields $y+x=x+y$.