Conditions That Imply Commutativity

Groups, Rings, Domains, Modules, etc, Galois theory
Post Reply
Tsakanikas Nickos
Community Team
Posts: 314
Joined: Tue Nov 10, 2015 8:25 pm

Conditions That Imply Commutativity

#1

Post by Tsakanikas Nickos »

Let \( \displaystyle R \) be an associative ring with unity \( \displaystyle 1_{R} \). Show that each of the following conditions imply that \( \displaystyle R \) is commutative:
  1. \( \displaystyle \forall r \in R : r^2 = r \)
  2. \( \displaystyle \forall r \in R : r^3 = r \)
  3. \( \displaystyle \forall r \in R : r^2 - r \in Z(R) \)
  4. \( \displaystyle \forall r \in R : r^2 + r \in Z(R) \)
  5. \( \displaystyle \forall r \in R : r^3 - r \in Z(R) \)
User avatar
Grigorios Kostakos
Founder
Founder
Posts: 461
Joined: Mon Nov 09, 2015 1:36 am
Location: Ioannina, Greece

Re: Conditions That Imply Commutativity

#2

Post by Grigorios Kostakos »

2. is answered here: Commutative ring
Grigorios Kostakos
Post Reply

Create an account or sign in to join the discussion

You need to be a member in order to post a reply

Create an account

Not a member? register to join our community
Members can start their own topics & subscribe to topics
It’s free and only takes a minute

Register

Sign in

Who is online

Users browsing this forum: No registered users and 11 guests