--- tags: mth350, dailyprep --- # Daily Prep 8B -- MTH 350-01 ## Overview In Module 8A we learned what a *ring* is and proved some basic properties about abstract rings. In Module 8B we will first of all wrap up any remaining concepts from 8A; then we'll take a closer look at *multiplicative inverses* in a ring. Unlike the case for additive inverses, not all elements in every ring have multiplicative inverses. So we'll look at more examples of rings and try to pick out those elements that do have them, paying special attention to rings in which *every nonzero* element has a multiplicative inverse. ## Learning objectives **Basic Learning Objectives:** *Before* our class meeting, use the Resources listed below to learn all of the following. You should be reasonably fluent with all of these tasks prior to our meeting; we will field questions on these, but they will not be retaught. - State the definition of a *multiplicative inverse* of a ring element. - State the definition of a *unit* in a ring. - Given a familiar ring, or a small finite abstract ring defined by operation tables, find all the elements that have multiplicative inverses (and then find the inverse of each one). **Advanced Learning Objectives:** *During and after* our class meeting, we will work on learning the following. Fluency with these is not required prior to class. - Give proofs for Theorems 7.10, 7.11, 7.15, and 7.16. - Identify the units of a ring that isn't one of the "familiar" ones. ## Resources for learning **Reading:** Read through Investigation 7 through the end of the proof of Theorem 7.16. *Note*: We'll be working through these proofs in class, so read through them in your prep, and try to fill in the missing reasoning. ## Exercises The exercises for this Daily Prep are found on the Google Form: https://docs.google.com/forms/d/e/1FAIpQLSfC03f5L1_PqFC6T9y7AUWmzGnT0akeqtAURrDxS6PJmGh8zA/viewform ## Submission and grading To submit your work, simply submit the Google Form. You will receive a receipt via email to confirm your submission. (Look in your spam folders if you do not see the receipt.) A **Pass** mark is given if the Daily Prep is turned in before its deadline and if each item on the Daily Prep has a response that represents a good faith effort to be right. **Mistakes are not penalized**. A **No Pass** is given if an item is left blank (even accidentally), has an answer but it shows insufficient effort (including responses like "I don't know"), or if the Daily Prep is late.