Hi everyone! This is my first post and it is more of an aside than a true post. In this post I will briefly discuss a few methods on how one creates a ring out of a set . Forgive me for being so brief, but this far from my area of math so I am less knowledgeable on the subject; however I found this rather interesting.
First we begin with the rather “stupid” method. If is finite we consider the bijection to . If is countable we consider the bijection to . This is rather uninteresting.
We can also construct the free group on the set , here we denote this . We can then look at the group ring .
Next, (forgive any abusive notation here) we can consider : the set of all polynomials with variables from where
We can also construct the power set and form a ring with addition being the symmetric difference and multiplication being the intersection.
Finally we can create , the exterior algebra on which is generated by things of the form where and with multiplication defined by the wedge product: with the relationship and . Thus . This will have basis when viewed as a vector space if is finite.