Note: R is a ring.
1. If A is an abelian group and n > 0 an integer such that na = 0, a A, then A is
a unitary Zn -module, with the action of Zn on A given by ka = ka, where k Z and
k k Zn under the canonical projection Z
III.6. Factorization in Polynomial Rings
1. (a) If D is an integral domain and c is an irreducible element in D, then D[x] is not a
principal ideal domain.
(b) Z[x] is not a principal ideal domain.
(c) If F is a eld and n 2, then F [x1
1. The set of all nilpotent elements in a commutative ring forms an ideal.
Proof: Let R be a commutative ring and let N denote the set of all nilpotent elements in R. Then
I = cfw_a R : for some n Z, rn = 0.
As 0 I , I =
Algebra Exam 1 Fall 2006
Instructions: In this exam, unless otherwise stated, G will denote a group, Z denotes the additive group
of integers, and Zm denotes the additive group of integers modulo m. Work on each of the problems.
1. Let G be an abelian gro
Homework Assignment 5
Homework 5. Due day: 11/6/06
(5A) Do each of the following.
(i) Compute the multiplication: (12)(16) in Z24 .
(ii) Determine the set of units in Z5 . Can we extend our conclusion on Z5 to Zp , for an arbitrary
prime integer p?
West Virginia University
Dr. Lizzie Santiago
College of Engineering and Mineral Resources
ENGR 293T: Engineering Process and Product
Tuesday and Thursday 2PM-2:50PM (CRN 88678)
Monday and Wednesday 3-3:50PM (CRN 88725)