This is an unformatted preview. Please download the attached document for the original format.Shawn Hicks California State University Fullerton Math 407 Edited by: Veronica Holbrook April 7 2014 Spring 2014 Solutions to Homework 7 Quotient Structures 1. If R is a ring and I is a subgroup of R (under addition) then I is an ideal if rx xr I for all r R and for all x I . Dene R/I = {r + I | r R}. Prove that R/I is a ring under the operations (r + I ) + (s + I ) = (r + s) + I and (r + I ) (s + I ) = rs + I for r s R. Recall that you must show that each operation is well-dened that is if you choose dierent representatives for the cosets then the sum and product do not change.