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Welcome to an Example MediaWiki Instance Hosted on Algebrist!
Consult the User's Guide for information on using the wiki software.
Getting started
- Configuration settings list
- MediaWiki FAQ
- MediaWiki release mailing list
- Localise MediaWiki for your language
- Learn how to combat spam on your wiki
Quick Start
MediaWiki takes the approach of not providing a real dashboard for editing. In the long run this makes things way easier but it can be a bit weird at the very start. For example if you look to a button to create a new page you wont find one. However, it is very easy to make new pages. Say you want to make a page called
FancyMathStuff
then you would, in the URL bar, enter
https://wiki.algebrist.ddns.net/index.php?title=FancyMathStuff
. On that page you will see the option to create it!
A lot of stuff in MediaWiki is like this, there's not a button for it but its actually quite easy. The docs are super helpful and well done.
Definition: An \(R\)-module \(M\) is a free \(R\)-module if \(M\) has a basis. A subset \(A\) of an \(R\)-module \(M\) is a basis of \(M\) if \(A\) is linearly independent and generates \(M\).
Examples:
- \(R=R\{1_R\}\) is a free \(R\)-module
- \(R^2=R\oplus R\) has basis \(\{(1_R,0_R),(0_R,1_R)\}\)
- \(R[x]\) is a free \(R\)-module with (infinite) basis \(\{1,x,x^2,...,x^i,...\}\)
- \(R[x,y]\) has basis \(\{x^n,y^m\ |\ n,m\geq 0 \}\) as free \(R\)-module
- \(R[x,y]\) has basis \(\{1,y,y^2, ...,y^i,...\}\) as free \(R[x]\)-module
All free \(R\)-modules \(M \) have some basis \(B=\{b_1,...,b_n\} \) so have rank \(n \), and can be written as \(R^n\cong M \). Additionally every element \(m\in M \) can be uniquely written \( m=\sum_{i=1}^n r_ib_i\).
