Homework Assignment 1

MATH 323: Rings and Modules
The University of British Columbia
Winter 2016 Term 2
Miljan Brakočević
Homework Assignment 1
This problem set is due on Thursday, January 14, 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.
Def. Let R be a ring. A subset R1 ⊂ R is called a subring if it is an additive subgroup of
R, closed under multiplication and 1 ∈ R1 .
Def. Let R be a ring. Note that it always has two so called trivial ideals: (0) and R. An
ideal I of R is called maximal if I 6= R and if J is an ideal containing I then J = I or J = R.
Do the Dummit & Foote (3rd ed) exercises:
Section 7.1: Exercises 6, 7, 13, 15
Section 7.3: Exercises 25, 29, 30
Section 7.4: Exercises 4, 5, 6
Do the following additional one:
A1. (a) Let R be an integral domain with finitely many elements. Prove that R is a field.
(b) Let p be a prime number. Prove that Z/pZ is a field.