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I have never seen, in any text, a ring whose multiplication is commutative being called an "abelian ring", even though this would make perfect sense, because this term would necessarily refer to multiplication (addition is commutative by definition, of course). Is there some historical reason for that? Did Abel, maybe, only study "additive" structures?

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Especially in french texts I often have seen the terminology of commutative groups. In my opinion this is far better than abelian groups, not only because it is consistent with the terminology of commutative rings, commutative monoids, commutative semigroups and commutative group schemes, but also since it is absurd to name such a universal notion by its inventor. Otherwise we would have to call groups, say, "Cayley-Kronecker-Galois-systems" ;-). –  Martin Brandenburg Jun 4 '14 at 10:43
@Martin Inventor? –  user1729 Jun 4 '14 at 10:48
@MartinBrandenburg Yes, I know. In English texts as well, commutative group appears very often, but so does abelian group. And I agree with you on your preference - but one can not oversee the fact that the word abelian has been used so much that it is now written with a lower-case letter. I know very few words that have managed that in mathematics. I am very intrigued to know why this happens. –  Bill_Werden Jun 4 '14 at 10:51
By the way: Another unnecessary inconsistency between group and ring theory is the following: The characteristic of a ring is the order of $1$ in the underlying additive group. However, there is one exception: Rings may have characteristic zero, but group elements have order $\infty$ instead. Again, in this case, I claim that the ring terminology is the better one. Orders shouldn't be infinite, but rather natural numbers (including $0$). But this is another subject ... –  Martin Brandenburg Jun 4 '14 at 10:51
Of course by "oversee" I mean "overlook". I keep forgetting that "up to equivalence" does not work in real life. –  Bill_Werden Jun 4 '14 at 11:00

3 Answers 3

To address your final sentence, "Did Abel, maybe, only study "additive" structures?": He studied algebraic equations which had commutative Galois group (independently of Galois).

Today commutative groups are generally called abelian, named after N. H. Abel, the famous Norwegian mathematician, who investigated a class of solvable algebraic equations related to commutative groups. - Fuch, Abelian groups, footnote in preface.

The relevant result of Abel is the following.

Theorem: If the roots of an equation of arbitrary degree are related among themselves in such a way that all the roots can be expressed rationally by means of one of them, which we denote by $x$; if in addition whenever one denotes by $\theta x$, $\theta_1x$ two other arbitrary roots, one has $$ \theta\theta_1x=\theta_1\theta x $$ then the equation to which they belong will always be solvable algebraically.

These equations were called Abelian equations by Kronecker, and they have abelian Galois group (thus the connection). Abelian groups were first called this by Jordan in 1870.

See Section 6.5 of the book Galois Theory by David Cox, 2004, for more details (both mathematical and historical) on these equations. See also the historical note on p42 of Fraleigh A first course in abstract algebra.

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+1. I think this is the correct answer: Abel actually studied commutative groups, but not commutative rings. –  user64687 Jun 4 '14 at 10:31

All rings are additive abelian groups, i.e. addition is commutative, but it may be the case that multiplication of the ring is not commutative. For example, the ring of $2\times 2$ matrices over $\mathbb{R}$ under ordinary matrix multiplication and addition is a ring, but its multiplication is not commutative. This definition was chosen as to encompass the most general set of examples while still providing enough structure to build up ring theory.

Edit: Francois Joseph Servois seems to be the first mathematician to use the term "commutative".

Source: http://mathforum.org/library/drmath/view/52599.html

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This is not the question. The OP is asking why groups are termed "abelian" and rings "commutative". This is only historical, and groups are also called "commutative". –  zarathustra Jun 4 '14 at 9:04
Oops misread. Sorry english is my second language. –  Samuel Cavazos Jun 4 '14 at 9:05
There we go. Added a source. –  Samuel Cavazos Jun 4 '14 at 9:14
Oh, thank you so very much for your useful answer and your even more useful edit. This clarifies everything. –  Bill_Werden Jun 4 '14 at 9:21
@Bill_Werden Did it really answer your question? It seems to me this answer is for the question "Why are not all rings commutative?" which is not the question you asked. –  Daniel Rust Jun 4 '14 at 9:23

Actually, Abelian groups can also be called commutative groups, and in some places authors call commutative rings abelian rings (or algebras). These usages are comparatively low, although it's understandable why they became somewhat interchangeable.

If you play around with the ngrams tool you'll also find that semigroups tend to be called commutative rather than abelian.

Tradition gave rise to the current common usage.

Incidentally, some other authors have begun to repurpose "abelian ring" to mean "a ring whose idempotents are central."

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I have certainly seen "commutative group", but never "abelian ring" (in your first sense). I would be curious to see an example! –  user64687 Jun 4 '14 at 10:30
@AsalBeagDubh in my experience, that usage is exceedingly rare and seemingly concentrated in works by authors who aren't really mathematicians. Hits like this are what I'm talking about. But actually "abelian algebra" has a measurable usage in real math literature. –  rschwieb Jun 4 '14 at 10:47
Google "abelian C*-algebra" or "abelian endomorphism ring" –  Martin Brandenburg Jun 4 '14 at 10:53
@rschwieb: thanks for the link! –  user64687 Jun 4 '14 at 10:54
I saw the term "abelian $C^*$-algebra" in Conway's text on functional analysis and it looks strange to me; "commutative $C^*$-algebra" looks more natural. –  KCd Jun 4 '14 at 11:07

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