Algebras, or When Can I Multiply Stuff in My Module?

Wordy Introduction, Motivation.

When you first start high school algebra, the big thing is FOIL-ing, right?  Factoring and factorizing quadratics.  When you get to calculus, the big things are derivatives and integrals.  Then when you get to college and start doing math, things get a little tougher.  We start learning about abstract structures, and these become increasingly specific and increasingly complex as we go along.

What the Hell is a Module?

This post is going to be a gentle introduction to what a module is.  It isn’t hard, but, for me, modules were sort of just “thrown in” with a whole bunch of defining properties and no motivation for why I should care about them.  I’m hoping to motivate them at least a little bit so that you feel more comfortable thinking and working with them!