--- tags: mth350, dailyprep --- # Daily Prep 12B -- MTH 350-01 ## Overview In our final lesson of the semester, we'll look at the formal definition of what it means for two rings to be *isomorphic* and think about how to prove that two rings are isomorphic; and use *isomorphism invariants* to show two rings are not isomorphic. ## 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 an *isomorphism* between two rings and what it means for two rings to be *isomorphic* to each other; use the definition to decide whether a specific function between two specific rings is an isomorphism and whether two specific rings are isomorphic. + State the steps necessary to prove that a two rings are isomorphic and that a function between two rings is an isomorphism; also state what it means to say that two rings are *not* isomorphic. **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. + Define the concept of an *isomorphism invariant* and give a list of isomorphism invariants. + Given two rings, decide whether they are isomorphic and then prove it by constructing an isomorphism between them. + Theorems to state and apply: Theorem 10.13. ## Resources for learning **Reading:** Re-read the section on "Isomorphisms of Rings", especially Definition 10.5; then read through the end of the Investigation. ## Exercises The exercises for this Daily Prep are found on the Google Form: https://docs.google.com/forms/d/e/1FAIpQLSd2FltfnrXeYHEAHHnwEqcScZEgpf76X_bheiybvXJeHTdbUQ/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.