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Preface A Note to Instructors

Welcome! Thank you for considering this text; it won't be for everyone, as strong opinions informed its creation. The strongest are (a) that students learn math best by doing it, and (b) that students–especially pre-service teachers–more naturally learn modern algebra by encountering rings first.

Pedagogically, these notes fall under the big tent of inquiry-based learning (IBL). Broadly, there are several types of statements you'll find as you read these notes.

  • Theorems: A numbered theorem is a statement that students are expected to prove for themselves. The authors generally assign 3–6 numbered theorems (or exercises, or lemmas) for each class meeting, with students expected to present their work during the next class. These presentations and the ensuing discussions form the regular work of the class. Students are not expected to prove unnumbered theorems. The unnumbered theorems unify nearby numbered theorems (such as stating an existence theorem and uniqueness theorem as a single result), or are otherwise too technical or complicated to be illuminating. Nonetheless, they are generally important results of which students should be aware.

  • Lemmas: There are a few lemmas in the notes. As a rule, these lemmas pull out a step from nearby theorems that might be too big to reasonably expect students to take by themselves. If you would like to suggest additional lemmas, feel free to get in touch with the authors.

  • Exercises: The exercises are generally computational in nature, and presage an upcoming generalization (or reinforce a recent theorem). They are generally labeled as Activity, Exploration, or Investigation. As such, more than a correct numerical answer is needed for a good solution to an exercise.

  • Challenges: There are a few (unnumbered) challenge problems in the text. These problems may be assigned or they may not, but they are generally difficult and their omission will not disrupt the flow of the text. Students may be interested merely in knowing their statements (e.g., \(\Z[x]\) is not a PID).

We begin with a brief overview of some results from elementary number theory regarding divisibility and primes, and introduce modular arithmetic. Other than induction, no proof techniques are explicitly discussed. It is assumed that students using these notes have had an introduction to proofs.

Brief attention is paid to fields before we dive in to rings. Other than mentioning their existence, no attention is given to noncommutative rings. Rings and ideals are developed with an eye toward eventually proving that every Euclidean domain is a unique factorization domain. We briefly explore nonunique factorization (though this could be done in outside homework, if desired) before turning to an exploration of homomorphisms and ideals in general.

As of this writing (July 2020), groups are not covered in this book. Depending on personal preference, with the time left at the end of the semester (often approximately 1–3 weeks, depending on your class's pace), you could present an introduction to groups directly to your students, or use freely available IBL material from the Journal of Inquiry-Based Learning in Mathematics.

The book has been used to carry a full semester course at least three times: twice at Dordt University (Fall 2018 and 2020), and once at Morningside College (Fall 2019). Future plans for the text include:

  • An expanded treatment of fields, with an emphasis on extensions of \(\Q[x]\) (and possible introduction to groups via permutations of roots of polynomials).
  • Additional optional end-of-section exercises.
  • Integration of SageMath cells to aid computation where appropriate.
  • PDF and HTML versions of both the student and instructor versions. These are available! Sort of. No HTML instructor version yet, but a PDF of the student version is now available.
  • Low-cost print copies of the student version. Coming soon!

There is no planned timeline for any of these projects. If you are interested in helping make one of these happen, please email me (Mike) and let me know! Or, if you just want to let me know that you've found the text useful, that would also be welcome news. And of course, if you find any typos or mistakes, I would love to know that as well.