Ultraproducts in commutative algebra

Since 1966, when Abraham Robinson [5] used ultrapowers to defend Liebniz' disreputed theory of infinitesimals in analysis, interest in ultrafilters and ultrapowers, in both analysis and algebra, has grown markedly. This paper will explore some general properties of ultra-powers of commutative rings.

Chapter 1 consists of the basic definitions and characteristics of filters, ultrafilters and ultraproducts. In particular, we will examine the conditions under which a given ring is isomorphic to its ultrapower.

In Chapter 2, we find necessary and sufficient conditions on a ring R under which certain ring-theoretic properties of R will be shared by its ultrapower.

The dimension of a commutative ring with identity is defined in Chapter 3, and the relationships between the dimension of a ring R and that of its ultrapower are discussed.

Chapter 4 states some results comparing two functors from R-modules to R_F-modules.

It is assumed that the reader has a working knowledge of com-mutative ring theory. Our basic reference for undefined terms will be Hungerford [4].