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