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I have a simple question answer to which would help me more deeply understand the concept of (non)commutative structures. Let's take for example (our teacher's definition of) a ring:

Let $R\neq \emptyset$ be a set, let $\oplus:A\times A \to A$ and $\bullet :A\times A \to A$ be binary operations. Moreover, let $(R, \oplus)$ be a commutative group, $(R, \bullet)$ be a monoid and following property holds for all $a, b, c\in R$: $$a\bullet(b\oplus c) = (a\bullet b)\oplus(a \bullet c)$$ $$(b\oplus c)\bullet a = (b\bullet a)\oplus(c \bullet a)$$ Then ordered triple $\mathbf R = (R, \oplus, \bullet \mathbf)$ is called a (unitary) ring.

Moreover, we call ring $\mathbf R$ commutative iff $(R, \bullet)$ is a commutative monoid. Commutativity of a ring is always a matter of its multiplicative operation because the additive operation is always assumed to be commutative.

Could anyone explain me the bold part? Why do we even in non-commutative rings (and fields etc.) assume the addition to be always commutative? Is there some serious reason? Would it make any trouble? Or studying of structures with non-commutative addition just doesn't give us anything new so we can take addition as commutative simply because of our comfort?

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3 Answers 3

up vote 6 down vote accepted

Perhaps the comment refers to the fact that in order to generalize rings to structures with noncommutative addiiton, one cannot simply delete the axiom that addition is commutative, since, in fact, other 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.

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

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 $\rm\,1\!:\ \ 1x = x$

In order to state this result concisely, 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

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Thank you, it's really interesting! Most books I have, talk about the topics just briefly without any deeper and detailed insight. These are the notes which I'd love to have my books full of... Accepted button goes to you. –  Jeyekomon Mar 31 '13 at 19:58

There are so-called near-semirings (http://en.wikipedia.org/wiki/Near-semiring) in which addition is non-commutative.

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Of course one can develop a theory in which the addition is not commutative (see Boris' answer and his mentioning the near-semirings).

Why "rings with non-commutative addition" are a somewhat side story and commutativity of addition is the usual assumption? Simply because the basic and main examples of these rings, those which primarily occur doing mathematics, do have this property.

I believe that by far most "rings" can be reconducted in a way or another to the ring of matrices over some algebraic structure with commutative addition (commutative rings or division algebras, tipically). Addition of such matrices commutes.

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