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I am vaguely familiar with the broad strokes of the development of group theory, first when ideas of geometric symmetries were studied in concrete settings without the abstract notion of a group available, and later as it was formalized by Cayley, Lagrange, etc (and later, infinite groups being well-developed). In any case, it's intuitively easy for me to imagine that there was substantial lay, scientific, and artistic interest in several of the concepts well-encoded by a theory of groups.

I know a few of the corresponding names for who developed the abstract formulation of rings initially (Wedderburn etc.), but I'm less aware of the ideas and problems that might have given rise to interest in ring structures. Of course, now they're terribly useful in lots of math, and $\mathbb{Z}$ is a natural model for elementary properties of commutative rings, and I'll wager number theorists had an interest in developing the concept. And if I wanted noncommutative models, matrices are a good place to start looking. But I'm not even familiar with what the state of knowledge and formalization of things like matrices/linear operators was at the time rings were developed, so maybe these aren't actually good examples for how rings might have been motivated.

Can anyone outline or point me to some basics on the history of the development of basic algebraic structures besides groups?

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Excellent question! I look forward to an answer! – BBischof Jul 21 '10 at 17:22
up vote 33 down vote accepted

For a nice introduction to the history of ring theory see the following paper

I. Kleiner. From numbers to rings: the early history of ring theory.
Elemente der Mathematik 53 (1998) 18-35.
SEALS: direct link to pdf, persistent link to article
Springerlink: direct link to pdf, persistent link to article

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Edit: Bill Dubuque has pointed out that much of this answer (specifically, the part about FLT) is essentially a mathematical urban legend, albeit a pervasive one. I cannot delete an accepted answer, so here is a link to an answer of his on MO explaining it.
Here is also a link to a related question.

There's some of the history here in Bourbaki's Commutative Algebra, in the appendix. Basically, a fair bit of ring theory was developed for algebraic number theory. This in turn was because people were trying to prove Fermat's last theorem.

Why's this? Let $p$ be a prime. Then the equation $x^p + y^p = z^p$ can be written as $\prod (x+\zeta_p^iy) = z^p$ for $\zeta_p$ a primitive $p$th root of unity. All these quantities are elements of the ring $Z[\zeta_p]$. So if $p>3$ and there is unique factorization in the ring $Z[\zeta_p]$, it isn't terribly hard to show that this is impossible at least in the basic case where $p $ does not divide $xyz$ (and can be found, for instance, in Borevich-Shafarevich's book on number theory).

Lame actually thought he had a proof of FLT via this argument. But he was wrong: these rings generally don't admit unique factorization. So, it became a problem to study these "generalized integers" $Z[\zeta_p]$, which of course are basic examples of rings. It wasn't until Dedekind that the right notion of unique factorization -- namely, factorization of ideals -- was found. In fact, the case of FLT I just mentioned generalizes to the case where $p$ does not divide the class number of $Z[\zeta_p]$ (the class number is the invariant that measures how far it is from being a UFD). And, according to this article, Dedekind was the first to define a ring.

The article I linked to, incidentally, has a fair bit of additional interesting history.

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How come you can write $x^p + y^p$ as $\prod (x+\zeta_p^i)$? – Casebash Jul 21 '10 at 21:11
Should have been $\prod (x + \zeta_p^i y)$, now fixed. This is because the two are polynomials (say in $y$ with $x$ fixed) with the same roots and same leading coefficient. – Akhil Mathew Jul 21 '10 at 21:25
It's a historical legend that work on FLT inspired much of number theory and ring theory. Rather, number theorists of that era were inspired by much loftier goals such as the pursuit of higher reciprocity laws. – Bill Dubuque Aug 6 '10 at 16:07
@Akhil: see e.g. my MO post Generally one should wary of historical remarks not made by serious historians. Popular accounts (e.g. E.T. Bell) include many romanticized legends. Note: I'm not a historian but, rather, a mathematician with a strong interest in history. – Bill Dubuque Aug 6 '10 at 19:35
Update Now there is an MO question on this precise topic – Bill Dubuque Aug 7 '10 at 19:26

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