Daily Prep 8A -- MTH 350-01

--- tags: mth350, dailyprep --- # Daily Prep 8A -- MTH 350-01 ## Overview Module 8 leads us to the heart of this course. Previous modules have been building up the idea that the integers with addition and multiplication satisfy various *axioms* that describe how arithmetic and algebra work in the integers; and also that many other "number" systems that also have some notion of "addition" and "multiplication" *also* satisfy many of those axioms. We've sought areas where all those systems agree with each other. Here in Module 8, we reach our final level of abstraction by defining the concept of a **ring**. A ring is an abstract system that consists of objects and two operations that satisfy the core of the arithmetic axioms for integers. But the precise nature of those objects and operations can be many different things. In Module 8A we'll define the concept of a ring precisely, then explore properties of the arithmetic and algebra in a ring. ## 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 *ring* and the seven *Ring Axioms*; use the Axioms and the definition to decide whether a set with two binary operations is or is not a ring. + State the definitions of a *commutative ring*, the *identity* of a ring and a *ring with identity*, and the *multiplicative inverse* of an element in a ring. + State the following results: Theorem 7.3, Theorem 7.5, Theorem 7.10, and Theorem 7.11. **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. + Given a set with addition and multiplication operations, decide whether the set forms a ring under the operations. If so, whether the ring is commutative; and whether there is an identity element. + Apply, and explain line-by-line the proofs of Theorem 7.3, Theorem 7.5, Theorem 7.10, and Theorem 7.11. ## Resources for learning **Reading:** Read through Investigation 7, up through and including Definition 7.13 (definition of a multiplicative inverse). ## Exercises The exercises for this Daily Prep are found on the Google Form: https://docs.google.com/forms/d/e/1FAIpQLSely9H05VuF5dWxMrOvwq1JRN-Ac5xnh0TUBllfiBxR7_NYAQ/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.