--- tags: mth350, dailyprep --- # Daily Prep 12A -- MTH 350-01 ## Overview In our final module in this course, we will take up the question of how to tell if two rings are "essentially the same" --- i.e. the rings have superficial differences but all the "essential" structural properties between the two are the same. We'll say that two such rings are **isomorphic** and develop a way of proving two rings are isomorphic using functions. ## 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. + (*Review*) Determine if a function between two sets is injective, surjective, or bijective. + State what it means for a function between two rings to *preserve an operation*. **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. + 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. ## Resources for learning **Reading:** Read Investigation 10 starting with the Preview Activity and continuing up to, but not including Activity 10.6. ## Exercises The exercises for this Daily Prep are found on the Google Form: https://docs.google.com/forms/d/e/1FAIpQLScpAowj-S2k4dDYOdUceNaLWsiRhVc-8vJSzugHz_SGxF6aFw/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.