# Rings, Fields, and Vector Spaces

Chapter
Part of the Mathematics in Mind book series (MATHMIN)

## Abstract

In this last stage of our exploration of mathematics, we will analyze three more algebraic structures of increasing complexity—rings, fields, and vector spaces. Herstein begins his chapter on rings in the following way:

As we indicated in Chapter 2, there are certain algebraic systems which serve as the building blocks for the structures comprising the subject which is today called modern algebra. At this stage of the development we have learned something about one of these, namely groups. It is our purpose now to introduce and to study a second such, namely rings. The abstract concept of a group has its origins in the set of mappings, or permutations, of a set onto itself. In contrast, rings stem from another and more familiar source, the set of integers. We shall see that they are patterned after, and are generalizations of, the algebraic aspects of the ordinary integers. (1975: 120)

## Bibliography

1. Herstein, I. (1975). Topics in Algebra. New York: John Wiley & Sons.
2. Johnson, M. (1987). The Body in the Mind. Chicago: University of Chicago Press.Google Scholar
3. Lakoff, G. & M. Turner. (1989). More Than Cool Reason: A Field Guide to Poetic Metaphor. Chicago: University of Chicago Press.
4. Lakoff, G. & R. Núñez. (2000). Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. New York: Basic Books.
5. Rosch, E. H. (1978). “Principles of categorization”. In: E. Rosch & B. Lloyd (eds.), Cognition and Categorization. Pages 27–48. Hillsdale, N.J.: Erlbaum Associates.Google Scholar
6. Talmy, L. (2000). Toward a Cognitive Semantics. Cambridge: The MIT Press.Google Scholar
7. Turner, M. (1996). The Literary Mind. Oxford & New York: Oxford University Press.Google Scholar
8. Van de Walle, J. (2007). Elementary and Middle School Mathematics Teaching Developmentally. Boston: Allyn and Bacon (Pearson).Google Scholar