A Theory And Development Of Quotient Rings

Abstract

The purpose of this paper is a logical and clear development of the concept of quotient rings. The development will begin with the concept of groups and quotient groups, followed by discussions of rings, modular quotient rings, polynomial quotient rings, and, finally, an extension to the quotient field of rational numbers. The desired result is a unified presentation of some of the more fundamental ideas of modern abstract algebra, with a strong emphasis on the quotient ring and topics which are necessary for its development. However, rather than beginning immediately with the desired topic of rings, it will be much more advantageous to first of all define the symbols and terms which will be used throughout the work, and then to consider some of the simpler and more basic ideas which will make the main topic much easier to understand.