Topics In Algebra

This book is intended as an introduction to that part of mathematics that today goes by the name of abstract algebra. The term abstract is a highly subjective one;' what is abstract to one person is very often concrete and down to earth to another, and vice versa. In relation to the current research activity in algebra, it could be described as not too abstract; from the point of view of someone schooled in the calculus and who is seeing the present material for the first time, it may very well be described as quite abstract. In ordinary English language the word "or" is implied to say that something is one or the other, that it is not both. The mathematical "or" is quite different, at least when we are speaking about set theory. For when we say that x is in A or x is in B we mean x is in at least on of A or B, and may be in both. One of the amazing features of twentieth century mathematics has been its recognition of the power of the abstract approach. This has given rise to a large body of new results and problems and has in fact led us to open up whole new areas of mathematics whose very existence had not even been suspected.