History of Ring Theory

Abstract

Algebra textbooks usually give the definition of a ring first and follow it with examples. Of course the examples came first, and the abstract definition later-much later. So we begin with examples. Among the most important examples of rings are the integers, polynomials, and matrices. "Simple" extensions of these examples are at the roots of ring theory. Specifically, we have in mind the following three examples: (a) The integers Z can be thought of as the appropriate subdomain of the field Q of rationals in which to do number theory. (The rationals themselves are unsuitable for that purpose: every rational is divisible by every other (nonzero) rational.) Take a simple extension field Q(α) of the rationals, where α is an algebraic number, that is, a root of a polynomial with integer coefficients. Q(α) is called an algebraic number field; it consists of polynomials in α with rational coefficients. For example, Q(√ 3) = {a + b √ 3 : a, b ∈ Q}. The appropriate subdomain of Q(α) in which to do number theory-the "integers" of Q(α)-consists of those elements that are roots of monic polynomials with integer coefficients, polynomials p(x) in which the coefficient of the highest power of x is 1. For example, the integers of Q(√ 3) are {a + b √ 3 : a, b ∈ Z} (this is not obvious). This is our first example. (b) The polynomial rings R[x] and R[x, y] in one and in two variables, respectively , share important properties but also differ in significant ways (R denotes the real numbers). In particular, while the roots of a polynomial in one variable constitute a discrete set of real numbers, the roots of a polynomial in two variables constitute a curve in the plane-a so-called algebraic curve. Our second example, then, is the ring of polynomials in two (or more) variables. (c) Square m × m matrices (for example, over the reals) can be viewed as m 2-tuples of real numbers with coordinate-wise addition and appropriate multiplication obeying the axioms of a ring. Our third example consists, more generally, of n-tuples R n of real numbers with coordinate-wise addition and appropriate multiplication, so that the resulting system is a (not necessarily commutative) ring. Such systems, often extensions of the complex numbers, were called in the nineteenth and early twentieth centuries hypercomplex number systems.