It is currently Tue Jun 18, 2019 10:10 pm


All times are UTC [ DST ]




Post new topic Reply to topic  [ 2 posts ] 
Author Message
 Post subject: Finite Boolean Ring
PostPosted: Sat Jun 25, 2016 6:59 am 
Team Member

Joined: Tue Nov 10, 2015 8:25 pm
Posts: 314
Here is a really difficult exercise!

Let \( \displaystyle R \) be a finite boolean ring. Show that there exists a set \( \displaystyle X \) and a ring isomorphism \( \displaystyle R \cong \mathcal{P}(X) \).



HINT
As \( \displaystyle X \) consider the set \( \displaystyle Hom\left( R, \mathbb{Z}_{2} \right) \) of ring homomorphisms \( \displaystyle R \rightarrow \mathbb{Z}_{2}\).


Top
Offline Profile  
Reply with quote  

 Post subject: Re: Finite Boolean Ring
PostPosted: Tue Aug 02, 2016 3:48 pm 
Team Member

Joined: Mon Nov 09, 2015 1:52 pm
Posts: 426
Hi Nickos.

Firstly, if \(\displaystyle{X}\) is a non-empty set, then, the power set \(\displaystyle{\mathbb{P}(X)}\)

is a \(\displaystyle{\rm{Boolean}}\) ring with addition

\(\displaystyle{A+B:=(A-B)\cup(B-A)\,,\forall\,A\,,B\in\mathbb{P}(X)}\)

and multiplication

\(\displaystyle{A\cdot B:=A\cap B\,,\forall\,A\,,B\in\mathbb{P}(X)}\)

and then \(\displaystyle{0_{\mathbb{P}(X)}=\varnothing\,\,\,,1_{\mathbb{P}(X)}=X}\).

Nickos, my idea is to define the map

\(\displaystyle{\Phi:R\to \mathbb{P}(\rm{Hom}(R,\mathbb{Z}_{2}))\,,r\mapsto \Phi(r)=\left\{f\in\rm{Hom}(R,\mathbb{Z}_{2}), f(r)=1\right\}}\)

which is a ring homomorphsim.

I am not sure if \(\displaystyle{\Phi}\) is one to one and onto.

I like very much this question and i gave an answer in order to make some progress.


Top
Offline Profile  
Reply with quote  

Display posts from previous:  Sort by  
Post new topic Reply to topic  [ 2 posts ] 

All times are UTC [ DST ]


Mathimatikoi Online

Users browsing this forum: No registered users and 1 guest


You cannot post new topics in this forum
You cannot reply to topics in this forum
You cannot edit your posts in this forum
You cannot delete your posts in this forum
You cannot post attachments in this forum

Search for:
Jump to:  
cron
Powered by phpBB® Forum Software © phpBB Group Color scheme created with Colorize It.
Theme created StylerBB.net