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MATH 323: Rings and Modules The University of British Columbia Winter 2015 Term 2 Miljan Brakočević Homework Assignment 6 This problem set is due on Thursday, February 25, 2016 at the end of class. Please staple your assignment. Late homework will not be accepted under any circumstances. E-mail submission will not be accepted. Reading Part I (mandatory): Please read the last 5 pages of the week 6 lectures PDF here: http://www.math.ubc.ca/~miljan/math323/notes/week6.pdf This amounts to: √ √ • Finishing example that we started today about the ring Z[ −5] and (1 + −5)(1 − √ √ √ −5) = 2 · 3 therein. All four elements 1 + −5, 1 − −5, 2, 3 are irreducible in this ring, nevertheless none of them is a prime. • A proposition: in a PID, being a prime element is equivalent to being an irreducible. • Its corollary: in a PID, every non-zero prime ideal is maximal. • The subsequent corollary about F[x]/(x) and the example about R[y]. Do the Dummit & Foote (3rd ed) exercises: Section 8.1: Exercises 7, 9, 10, 11 Section 8.2: Exercises 1, 3, 5 Section 8.3: Exercise 8 - - - - - - - - The above concludes the midterm material - - - - - - - -. Reading Part II (optional): Please read the following in Dummit & Foote proving that ring Z[(1 + yet not an Euclidean domain: √ −19)/2] is a PID • The example on page 277 together with immediately preceeding paragraphs. • The example on page 282 together with immediately preceeding paragraphs. 1