Chapter 3

Rings

Algebra is the intellectual instrument which has been created for rendering clear the quantitative aspect of the world.

—Alfred North Whitehead

Mathematics takes us still further from what is human into the region of absolute necessity, to which not only the actual world, but every possible world must conform.

—Bertrand Russell

Two of the earliest sources of the theory of rings lie in geometry and number theory. The study of surfaces determined by polynomial equations involved the addition and multiplication of polynomials in several variables. In addition, attempts to extend the prime factorization theorem for integers led to consideration of sets of complex numbers that were closed under addition and multiplication. Both cases involve a commutative multiplication. David Hilbert, who coined the term ring, and Richard Dedekind began the abstraction of these systems.

Earlier, in 1843, William Rowan Hamilton had introduced his quaternions. They are a noncommutative ring that contains the complex numbers, and he developed a calculus for them that he hoped would be useful in physics. At about the same time, Hermann Günther Grassmann was studying rings obtained by introducing a multiplication in what would today be called a finite dimensional vector space. The study of these “ hypercomplex numbers” culminated in 1909 in the structure theorems of Joseph Henry MacLagan Wedderburn, which mark the beginning of noncommutative ring theory.

However, it was not until 1921 that ...