October 24, 2020

Focus on an Algebraic

These notes have two major parts: in one, we focus on an algebraic structure called a group; in the other, we focus on a special kind of group, a ring. In the first semester, therefore, we want to cover Chapters 2–5. Since a rigorous approach requires some introduction, we review some basics of the integers and the natural numbers – but only to solidify the foundation of what students have already learned; we do not delve into number theory proper. Some of these ideas fit well with monomials, which I study “on occasion.” In algebra, a boring discussion of integers and monomials naturally turns into a fascinating story of the basics of monoids, which makes for a gentle introduction to groups. I yielded to temptation and threw that in. That should make life easier later on, anyway; a glance at monoids, focusing on commutative monoids without requiring commutativity, allows us to introduce prized ideas that will be developed in much more depth with groups, only in a context with which students are far more familiar. Repetitio mater studiorum, and all that.3 We restrict ourselves to the more comfortable notions since the point is to get to groups, and quickly.

A two-semester sequence on modern algebra ought to introduce students to the fundamental aspects of groups and rings. That’s already a bit more than most can chew, and I have difficulty covering even the stuff I think is necessary. Unfortunately, most every algebra text I’ve encountered expend far too much effort in the first 50–100 pages with material that is not algebra. The usual culprit is number theory, but it is by no means the sole offender. Who has that kind of time? Then there’s the whole argument about whether to start with groups, rings, semigroups, or monoids. Desiring a mix of simplicity and utility, I decided to write out some notes that would get me into groups as soon as possible. Voilà. You still haven’t explained why it looks like a textbook. That’s because I wanted to organize, edit, rearrange, modify, and extend my notes quickly. I also wanted them in digital form, so that (a) I could read them, two and (b) I’d be less likely to lose them. I used a software program called Lyx, which builds on LATEX; see the Acknowledgments. What if I’d prefer an actual textbook? but you can use this website for notation conversion.

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