A simple domain not being skew field?

  • Thread starter Thread starter peteryellow
  • Start date Start date
  • Tags Tags
    Domain Field
peteryellow
Messages
47
Reaction score
0
Can you find an example of a simple domain not being skew field?
 
Physics news on Phys.org
Like in your other thread: What are your thoughts on this? Can you list examples of simple rings? Domains?

What if we assume commutativity, i.e. can you find a simple integral domain that isn't a field? You shouldn't - but this might shed some light on the noncommutative case.
 
Simple ring M_n(F) where F is a field, but what is the definition of a simple domain?
 
A domain is a ring without zero divisors, i.e. xy=0 implies either x=0 or y=0. A simple domain is a domain that is a simple ring.

Unfortunately M_n(F) isn't a domain, so you're going to have to be more creative if you want to come up with an example!
 

Thread 'Trouble understanding an online solution to an exercise in Dummit & Foote'

##\textbf{Exercise 10}:## I came across the following solution online: Questions: 1. When the author states in "that ring (not sure if he is referring to ##R## or ##R/\mathfrak{p}##, but I am guessing the later) ##x_n x_{n+1}=0## for all odd $n$ and ##x_{n+1}## is invertible, so that ##x_n=0##" 2. How does ##x_nx_{n+1}=0## implies that ##x_{n+1}## is invertible and ##x_n=0##. I mean if the quotient ring ##R/\mathfrak{p}## is an integral domain, and ##x_{n+1}## is invertible then...
View full post »

Thread 'Questions about non existence of GCDs vs (coimages, cokernels)'

The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
View full post »

Thread 'Decomposition into irreps of compact Lie group'

When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
View full post »

Similar threads

I Un-skewing a skew symmetric matrix (for want of a better phrase)
Replies
1
Views
2K
I Localising a non integral domain at a prime
Replies
17
Views
1K
A Bivariate Smoothing Splines
Replies
1
Views
2K
B The Relationship Between Function and Ordering Relation
Replies
8
Views
1K
I Meaning of "identify" & variations in different math context
Replies
5
Views
514
MHB Definition of a Euclidean Domain ....
Replies
2
Views
2K
I Want to understand how to express the derivative as a matrix
Replies
8
Views
2K
A Reason out the cross product (for the moment): a skew symmetric form
Replies
14
Views
3K
MHB Principal Ideal Domains .... ....
Replies
2
Views
1K
Vectors as geometric objects and vectors as any mathematical objects
Replies
27
Views
2K

Hot Threads

Recent Insights

Back
Top