advertisement

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. 1