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Summary

This chapter contains a preliminary discussion of the aims and philosophy of the book, and is not logically part of the course. Some of the material may be harder to follow here than when it is treated more formally later in the book, so if you get stuck on something, don't worry too much, just skip on to the next item.

Where we're going

The purpose of this course is to build one of the bridges between algebra and geometry. Not the Erlangen program (linking geometries via transformation groups with abstract group theory) but a quite different bridge linking rings A and geometric objects X; the basic idea is that it is often possible to view a ring A as a certain ring of functions on a space X, to recover X as the set of maximal or prime ideals of A, and to derive pleasure and profit from the two-way traffic between the different worlds on each side.

Algebra here means rings, always commutative with a 1, and usually closely related to a polynomial ring k[x₁,…, x_n] or ℤ[x₁, …, x_n] over a field k or the integers ℤ, or a ring obtained from one of these by taking a quotient by an ideal, a ring of fractions, a power series completion, and so on; also their ideals and modules. In this book, A usually stands for a ring and k for a field, and I sometimes use these notations without comment.