Why are groups "abelian" but rings "commutative"?

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?

Solution 1:

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.

Solution 2:

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."